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[Mu]⁺SR measurement of the temperature dependence of the magnetic penetration depth in YBa₂Cu₃O₆ 95 single… Sonier, Jeffrey E. 1994

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/i+SR MEASUREMENT OF THE TEMPERATURE D E P E N D E N C E OF THE MAGNETIC PENETRATION D E P T H IN YBa2Cu30695 SINGLE CRYSTALS By Jeffrey E. Sonier Hon. B.Sc. Univ. of Western Ont., 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © Jeffrey E. Sonier, 1994 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics The University of British Columbia 6224 Agricultural Road Vancouver, Canada V6T 1W5 Date: y-( 3-7, iin A b s t r a c t This thesis presents the results of transverse-field muon spin rotation (TF-/Li+SR) mea-surements of the magnetic penetration depth in the a6-plane (i.e. Aa&), for the vortex state of high quality single crystals of YBaaCusOg.gs. In particular, the low-temperature dependence of Aa(, was determined in an effort to clarify the nature of the pairing mech-anism in the YBa2Cu306.95 compound. These results should be more reliable than previous jU+SR studies on powders and crystal mosaics, due to the employment of a novel low-background apparatus, as well as to improvements in sample quality and in the fitting procedure. A strong linear temperature dependence for 1/A^, was found to exist below 50K for applied magnetic fields of 0.5T and 1.5T. This linear temperature dependence contra-dicts the consensus of previous / i + SR studies which suggested a behaviour consistent with conventional s-wave pairing of carriers in the superconducting state. The presence of a linear term in the data reported here, supports recent microwave cavity measure-ments in zero field and indicates the existence of a more unconventional pairing state. In addition, a possible field dependence for Aa& at low temperatures was indicated by the data, with Xab(T = 0) having a range of 1347-145 lA and 1437-1496A for the 0.5T and 1.5T data , respectively. The range of these values was determined by fitting the data several different ways. For each type of analysis, Aof,(0) was found to be greater and the linear term was stronger in the 0.5T data. Furthermore, the 1.5T data appear to agree better with the microwave cavity measurements. Included in this thesis is a qualitative description of the conventional s-wave pairing state and a proposed <i-wave pairing state, called dx2_y2. The findings in this ^ + S R ii study support the latter, but does not rule out the possibility of other anisotropic pairing states or isotropic pairing theories in which critical fluctuations persist down to very low temperatures. 111 Table of Contents Abs trac t ii Table of Content s iv List of Tables vi List of Figures vii A c k n o w l e d g e m e n t s x 1 In troduct ion 1 2 T h e o r y 5 2.1 Magnetic Properties of Conventional Superconductors 5 2.1.1 Type-I Superconductors 5 2.1.2 The London Penetration Depth 5 2.1.3 The Coherence Length (Pippard 's Equation) . . . . . . . . . . . 7 2.1.4 Ginzburg-Landau Theory 9 2.1.5 The Clean and Dirty Limits 11 2.1.6 BCS Theory for Conventional Superconductors 12 2.1.7 Type-II Superconductors and the Vortex State . . . . . . . . . . 17 2.2 The Pairing Mechanism 24 2.2.1 5-Wave Pairing 24 2.2.2 The Motivation for an Alternative Pairing Mechanism 30 iv 2.2.3 d-Wave Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Anisotropy in YBa2Cu307_6 . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . 46 2.3.2 Anisotropy of the Magnetic Penetration Depth . . . . . . . . . . 46 2.3.3 Anisotropy of the Energy Gap . . . . . . . . . . . . . . . . . . . 48 3 Measur ing t h e p e n e t r a t i o n d e p t h w i t h T F - / i + S R 50 3.1 The Field Distribution 50 3.2 The Role of the Positive Muon . 64 3.3 The Raw Asymmetry For 2-Counter Geometry 68 3.4 The Corrected Asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.5 The Relaxation Function 71 3.6 Modelling The Asymmetry Spectrum of the Vortex State 74 3.7 4-Counter Geometry and the Complex Polarization . . . . . . . . . . . . 75 3.8 The Rotating Reference Frame 78 4 E x p e r i m e n t a l D e t e r m i n a t i o n of \ab(T) 79 4.1 Sample Characteristics 79 4.2 The Apparatus 79 4.3 The Measured Asymmetry . 82 4.4 Data Analysis 88 5 Conclus ion 120 Bibl iography 123 A S o m e R e m a r k s A b o u t T h e Fi t t ing P r o g r a m 128 v List of T a b l e s 4.1 Comparison of the fitting procedures. . . . . . . . . . . . . . . . . . . . . 119 A.l Accuracy of the Taylor series expansion used in the fitting program. . . 129 VI List of F i g u r e s 2.1 Magnetization Curve for a Type-I Superconductor 6 2.2 Field Penetration at the Surface of a Semi-Infinite Superconductor . . . 8 2.3 Energy Gap for an Isotropic s-Wave Superconductor 16 2.4 Magnetization Curve for a Type-II Superconductor 18 2.5 Vortex State of a Type-II Superconductor 20 2.6 The Normal Ground State of a Metal . . . . . . . . . . . . . . . . . . . 26 2.7 The Ground State of a BCS Superconductor . . . . . . . . . . . . . . . . 29 2.8 Excited State of a BCS Superconductor 31 2.9 Crystallographic Structure of YBa2Cu307_^ 34 2.10 Phase Diagram for Y B a 2 C u 3 0 7 _ 5 35 2.11 dx2_y2-WsLve Symmetry 38 2.12 <ia;2_2/2-Wave Symmetry in 3-Dimensional fc-Space 39 2.13 Antiferromagnetic Square Lattice 40 2.14 Scattering of Electron Pairs 42 2.15 Identifying the Nodes in the Energy Gap 44 3.1 A Perfect Triangular Flux Lattice 51 3.2 Theoretical Field Profile 56 3.3 Dependence of the Field Profile on A . 57 3.4 Dependence of the Field Profile on K . . . . . . . . . . . . . . . . . . . . 58 3.5 Theoretical Supercurrent Density Profile 59 3.6 Dependence of the Supercurrent Density Profile on A . . . . . . . . . . . 60 vii 3.7 Dependence of the Supercurrent Density Profile on K . . . . . . . . . . . 61 3.8 Dependence of the Average Supercurrent Density on Magnetic Field . . 62 3.9 A Perfect Triangular Flux Lattice 63 3.10 Transverse-Field Muon Spin Rotation Arrangement . . . . . . . . . . . . 66 3.11 2-Counter Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.12 Sample Asymmetry Spectra 73 3.13 4-Counter Geometry 76 4.1 Experimental Setup 81 4.2 The Asymmetry Spectrum in the Normal State . . . . . . . . . . . . . . 84 4.3 The Temperature Dependence of the Asymmetry Spectrum . . . . . . . 85 4.4 The Temperature Dependence of the Frequency Distribution . . . . . . 86 4.5 Changing the Field Below Tc 89 4.6 Single Gaussian Fits . 91 4.7 Linewidth Competition 93 4.8 Correlation of The Linewidth Parameters 94 4.9 x2 from Global Fitting 95 4.10 Background Signal Fitt ing Parameters 98 4.11 Sample Signal Fitting Parameters 100 4.12 1/X2ab vs. T f o r 0.5T applied field , 102 4.13 1/X2ab vs. T for 0.5T and 1.5T applied fields 104 4.14 Comparison with microwave results 105 4.15 Background Signal Fitting Parameters 107 4.16 Sample Signal Fitting Parameters 108 4.17 1/Xflb vs. T f o r 0.5T and 1.5T applied fields . 1 1 0 4.18 Comparison with microwave results I l l vm 4.19 Background Signal Fitt ing Parameters . . . . . . . . . . 4.20 Sample Signal Fitt ing Parameters . . . . . . . . . . . . . 4.21 l/\2ab vs. T for 0.5T and 1.5T applied fields . . . . . . 4.22 l/\2ab vs. T for 0.5T and 1.5T applied fields (full scan) 4.23 Comparison with microwave results . . . . . . . . . . . 4.24 The total asymmetry amplitude . IX A c k n o w l e d g e m e n t s I would like to thank my thesis advisor Rob Kiefl for many patient and informative discussions over the past two years, all of which have accelerated my research and heightened my interest in the field of high-Tc superconductors. Equal thanks to Jess Brewer, the TRIUMF spokesperson for this experiment and the author of many of the valuable computer programs used in the analysis process. Additional thanks to my predecessor in this work, Tanya Riseman, who provided me with the algorithm which forms the basis for the computer program I used to fit the data. I am also appreciative of the early help from Juerg Schneider, who helped design the low-background apparatus used in this experiment. For making the won-derful YBa2Cu306.95 crystals, I thank Ruixing Liang. For running shifts in this exper-iment and /or participating in inspirational discussions, I would like to thank Walter Hardy, Doug Bonn, Andrew MacFarlane, Gerald Morris, Kim Chow, Tim Duty, Phillipe Mendels, J im Carolan and Arnold Sikkema. Also, thanks to Curtis Ballard and Keith Hoyle for their technical assistance. x Chapter 1 In troduct ion In recent years, "high-Tc" superconductors have generated a frenzy of experimental and theoretical endeavours, motivated by the likelihood of spawning revolutionary technolo-gies. Their mere existence challenges our present-day understanding of many facets of condensed matter physics. The particular features which distinguish this class of mate-rials from conventional BCS superconductors have been well documented. Besides the extraordinarily high transition temperatures in excess of 120K, some of the more note-able differences include: anomalous normal-state properties, strong anisotropy, a very small coherence length and unconventional behaviour in the superconducting state. Early experiments intended to answer some of the broader questions concerning the nature of the mechanism forming the basis for superconductivity in the high-Tc materials, yielded conflicting and questionable results. More recently, the experiments have become more reliable, mainly because of vast improvements in sample quality. Researchers now recognize that the inherent differences between high-Tc powders, thin films and single crystals, are significant enough that they must be accounted for when interpreting and comparing results. Furthermore, it is now understood that impurities play a major role in the outcome of many of these experiments, so that often only experiments done with the same sample are comparable. Theories regarding the high-Tc compounds are divided into two major categories: "Fermi liquid" and "non-Fermi liquid" theories. The normal state of a conventional 1 Chapter 1. Introduction 2 BCS superconductor is well described by a Fermi liquid. However, the anomalous normal-state properties of the high-Tc compounds hint at the possible absence of a Fermi surface in this class of materials. The interpretation of experimental measure-ments is dependent upon which class of theories one assumes. For now at least, there is an unwillingness to stray too far from conventional thinking and the successes of BCS theory, so that experimental results are usually interpreted in the context of Fermi liquid theories. With the general assumption that the electronic ground state of the high-Tc su-perconductors is composed of paired charge carriers, determining the symmetry of the paring state has been the subject of much controversy. Despite many of the earlier re-ports, it is now generally agreed that conventional s-wave pairing of carriers is unlikely. The pairing state is almost certainly anisotropic, but beyond this, little is known for certain. A leading candidate in the high-Tc materials is the d-wave form of pairing called dx2__y2 [1,2]. For dx2_y2 symmetry, the wave-vector dependence of the energy gap which develops at the Fermi surface in the superconducting state is such tha t line nodes appear in the gap. Measurements of the temperature dependence of the magnetic penetration depth A are one way to probe the nature of the low-energy excitations and the symmetry of the pairing state. For an isotropic s-wave superconductor, the magnetic penetration depth A(T) varies exponentially at low temperatures. Even if the superconducting gap is anisotropic, as long as it does not vanish anywhere on the Fermi surface, the penetration depth A(T) will still exhibit exponential behaviour at temperatures such that ksT is less than the minimum value of the energy gap for excitations [3]. However, if the symmetry of the superconducting state is dx2_y2, with the asso-ciated line nodes running along the cylindrical Fermi surface, Cooper pairs can be broken more easily and A(T) is expected to change linearly with temperature at low Chapter 1. Introduction 3 temperatures [4,5]. Recent microwave cavity perturbat ion measurements of A A (i.e. X(T) — A(0)) on high quality YBa2Cu306.95 single crystals show a strong linear term below 30A" [6]. Studies of Tl2CaBa2Cu208-5 single crystals also show a linear de-pendence on T for A(T) at low temperatures [7]. However, previous experiments in Bi 2Sr 2CaCu 208 and YBa2Cu307_j thin films showed a T2 dependence on A(T) [8]. It has been pointed out that impurity scattering in a dx2_y2-WEwe superconductor can change the low-temperature behaviour from A(T) ~ T to A(T) ~ T2 [5,6,7]. Muon spin rotation is the most direct way to measure Aaj (the penetration depth in the a6-plane) in the bulk of the sample. Unlike other techniques, which are performed in zero-static magnetic field, / i + SR directly measures the magnetic field distribution associated with the vortex lattice of a type II superconductor. The penetration depth Aaj, is the length scale over which magnetic flux leaks into the superconducting regions around the vortex cores. The ^ + S R technique provides a means by which one can investigate not only the temperature dependence of \ab, but also its field dependence. The difficulty in the /U+SR technique lies in the extraction of Xab from the measured data. Here, one relies heavily on a phenomenological model of the magnetic field distribution and the ability of the fitting program to extract the relevant parameters. Previous /^+SR studies on sintered powders and crystals of YBa2Cu307_5, which were of lower quality than the ones used in this study, provide evidence in support of an isotropic s-wave pairing state [9]. However, impurities or other crystalline imperfec-tions could lead to a T 2 dependence. Furthermore, conclusions regarding the pairing mechanism were based on the overall behaviour of A(T) rather than that in the low-temperature regime. It has been argued that the temperature dependence of A(T) at higher temperatures can be drastically altered by strong-coupling corrections, impurity scattering, the precise shape of the Fermi surface and gap anisotropy [4]. A universal equation for A(T) at all T < Tc, based on the d-wave formalism, has yet Chapter 1. Introduction 4 to be determined. One must rely solely on the low-temperature deviations from s-wave theory, anticipated from purely qualitative arguments. Consequently, conclusions with regard to the pairing mechanism in YBa2Cus07_5 requires precise low temperature data. Previous ^ + S R measurements of X(T) either lacked good low-temperature da ta or were done on sintered powders. Low-temperature measurements of X(T) should also be able to reconcile the nature of the nodes in the gap, if they do indeed exist. That is to say, point nodes on the Fermi surface should be distinguishable from the line nodes associated with dx2_yi pairing [4]. This present study is concerned with recent /U+SR measurements of A„f, in single crystals of YB^CusOe.gs- Some words of caution must be provided to the reader. Recent discovery of a significant a-b anisotropy in the magnetic penetration depth [10] may have a dramatic influence on the interpretation of measurements of Aaj(T). Although the existence of a-b anisotropy is acknowledged in this study, it is not included in the final analysis, as the mat ter is presently under investigation. Also, during the final stages of this thesis there have been recent suggestions that X{T) is controlled by critical fluctuations down to very low temperatures [11]. In this case, the measured X(T) has little to do with the pairing mechanism, but is determined by the 3D XY and 2D XY universality classes. This possibility is not discussed any further in this thesis. Chapter 2 T h e o r y 2.1 M a g n e t i c Proper t i e s of Convent ional Superconductors 2.1.1 T y p e - I S u p e r c o n d u c t o r s The Meissner effect, which has long been used to characterize conventional supercon-ductors, is also readily observed in high temperature superconductors. When these compounds are cooled below a critical temperature Tc in the presence of an externally applied magnetic field, all of the magnetic flux is expelled from their interior in the absence of flux pinning. When the external field exceeds a critical value (Hc), the super-conductor returns to its normal state. Superconductors which obey the magnetization curve depicted in Fig. 2.1 are called type-I superconductors. The critical field HC(T) which separates the superconducting and normal phases is a temperature dependent quantity. 2.1.2 T h e L o n d o n P e n e t r a t i o n D e p t h In 1935, F . and H. London modified an essential equation of electrodynamics (i.e. Ohm's Law) in such a way as to obtain the Meissner effect, without altering Maxwell's equations themselves. In doing so, they incorporated the two-fluid model of Gorter and Casimir [12]. The two-fluid model separates the electron system into a superconducting component with an electron density ns, and a normal component with an electron density nn. They assumed the total electron density na = ns + nn behaved such that 5 Chapter 2. Theory 6 M I ^ MoHc /x0H Full Meissner Normal State Figure 2.1: Magnetization curve for a type-I superconductor. M is the average mag-netization defined by B = [i0H + M , where B is the spatial average of the field in the superconductor. 7^g ^ TIQ ctS T —> 0, and ran = n0 when T > Tc. Emerging from their phenomenological model is the so-called London penetration depth XL [13]: 1 47re2n, Ai m*c2 (2.1) where m* is the effective mass of the superconducting carriers, and ns is the supercon-ductor carrier density. Pure-elemental metals and low concentration alloys [eg. dilute Snln alloy system [14]), where m* fa me (free electron mass), tend to be type-I. Gener-ally speaking, A^ is short and the Fermi velocity of the superconducting carriers vp is high in these materials [15]. The physical significance of Ajr, pertains to the failure of the Meissner effect to occur abruptly at the surface of a superconductor; rather the magnetic field penetrates slightly into the bulk of the superconducting material on a length scale given by A^. Consider a semi-infinite superconductor with the boundary between normal and superconducting Chapter 2. Theory 7 regions at x = 0. In the presence of an external magnetic field applied perpendicular to the surface, London theory predicts that the magnetic flux decays exponentially into the bulk of the superconductor according to [13,16]: B(x) = B(0)e'x/XL (2.2) where B(0) is the magnetic field at the surface of the superconductor, and x is the distance into the superconductor from the surface (see Fig. 2.2). The expulsion of the magnetic field is accomplished by shielding currents (or supercurrents) which flow at the surface of the superconductor without any Ohmic losses. These supercurrents form a finite sheath which is spread out to a thickness A^ into the sample. Such a distribution of current is much more energetically favourable than an infinitely thin —* sheath of current [13]. The supercurrent density Js is related to the local magnetic flux density B by Ampere's Law: V x B = fiofs (2.3) 2.1.3 T h e Coherence Length (P ippard's Equat ion) The London equations whose derivation appears elsewhere (see [13] for instance), are —* local equations (i.e. they relate the current density at a point r*to the vector potential A at the same point), and hence define the superconducting properties as such. However, early discrepancies between experimental estimations of A L ( 0 ) (the penetration depth at zero temperature) for certain conventional superconductors, and those predicted by Eq. (2.1) led Pippard [13,15] in 1950 to introduce non-local effects into the London equations. Spatial changes of quantities such as ns in a superconductor may only occur on a finite length scale, the coherence length £0, and not over arbitrarily small distances. That is to say, whenever ns is varying in space its value may change significantly over Chapter 2. Theory 8 x=0 Figure 2.2: The magnetic field B0 at the surface of the superconductor decays to B0/e at a distance x = A in the interior of the superconductor (shaded region). distances of order £0- Thus the coherence length defines the intrinsic nonlocality of the superconducting state. Using the uncertainty principle, P ippard estimated the coherence length for a pure metal to be [17,18]: Co - 0.18- (2.4) TTA(O) kBTc where ks is Boltzmann's constant, and A(0) is the energy gap that forms at the Fermi surface in the superconducting state at absolute zero. This will be discussed in more detail later. In the London model, the local value of the magnetic flux density B{f) and the supercurrent density J(f) are assumed to vary slowly in space on a length scale A/,, and have negligible variation over distances ~ £0. In Fig. 2.2, the superconducting carrier density ns jumps discontinuously from zero to a maximum value at the sample surface. As already mentioned, ns is expected to make a significant change like this only over a distance on the order of £0. Thus in Fig. 2.2, £D = 0, so that one would Chapter 2. Theory 9 anticipate an exponential screening of the magnetic field in a real superconductor only when \L >> £0- Indeed, the London model is valid only for A^ >> £0. In a type-I superconductor, the magnetic penetration depth A/, is much less than the coherence length £0, so that ns does not reach its maximum value near the surface. That is to say, not all of the electrons within a thickness £0 from the surface of the superconductor contribute to the screening currents. Consequently, the penetration of the magnetic field does not follow the exponential form of Eq. (2.2) until one ventures an appreciable distance x ~ £0 into the superconductor. Thus the usual London model is inadequate in describing type-I superconductors where, £0 ^> X^. A modification of the London equations which gives the magnetic flux and supercurrent densities a more rapid variation in space, predicts the actual penetration depth A in a type-I superconductor to be [14]: A = (0.62A^ o) 1 / 3 , (£0 > \L) (2.5) where A is larger than A/,, but much smaller than £„. 2.1.4 Ginzburg-Landau T h e o r y In 1950, Ginzburg and Landau introduced the so-called superconducting order -param-eter ty(r) to describe the nonlocality of the superconducting properties. \l/(r*) can be thought of as a measure of the order in the superconducting state at position f below Tc. It has the properties that * ( r ) -> 0 as T -> Tc, and | # ( f ) | 2 = ns(f) (i.e. the local density of superconducting electrons). At the time Ginzburg-Landau (GL) theory was developed, the nature of the superconducting carriers was yet to be determined. Inter-preting m* as the effective mass and q as the charge of the fundamental superconducting Chapter 2. Theory 10 particles (whatever they may be), they derived the penetration depth as: A(T) = - ^ L - (2.6) 47rg2 |*0 |2 where | ^ 0 | 2 is the value of |\I/|2 deep inside the superconductor (i.e. its equilibrium value). The coherence length defined in the GL-formalism is: t(T) # (27) \2m*\a(T)\ K ' where a(T) is a temperature-dependent coefficient in the series expansion of the free energy (see [19]). As written, (2.6) and (2.7) suggest that the penetration depth A(T) and the coherence length £(T) are temperature dependent quantities. Eq. (2.7) is closely related to the Pippard coherence length £• defined in Eq. (2.4). In GL-theory, the coherence length £(T) is the characteristic length for variations in |\l/(f)|2 = ns(r). One noteable difference is that Pippard 's coherence length £0 is independent of tem-perature. In fact in the dirty limit £(T = 0) reduces to Eq. (2.4). Near the transition temperature Tc both A(T) and £(T) vary as (1 — T / T c ) - 1 ' 2 , prompting the introduction of the Ginzburg-Landau parameter K, where: - m (2-8) The variation of A(T) and ^(T) with temperture near Tc comes from an expansion of the free energy density to first-order in the field H. It follows that K in (2.8) is temper-ature independent to this order. An exact calculation from microscopic theory gives a weak temperature dependence for /c, with K increasing for decreasing temperature T. Through energy considerations, Ginzburg and Landau characterized a type-I super-conductor as one in which K < l / v 2 . For K >> 1, GL theory reduces to the London model. Chapter 2. Theory 11 2.1.5 T h e Clean and Dirty Limits The purity of a superconductor is characterized by the ratio Z/£0- £o is the coherence length of the pure material, and is given by Eq. (2.4). / is the electron mean free path, defined as: / = TVF (2.9) where r is the time interval between collisions of conduction electrons with impurities in the sample. The magnitude of r is determined in the normal state. A sample is clean if //£<, ^> 1, and dirty if l/£0 <C 1 [20]. The actual coherence length when impurities are considered, is dependent upon the mean free pa th /. Intuitively, one can define an effective coherence length £(/) such that [13,17]: m=i+T (2-10) According to Eq. (2.10), the coherence length £(/) becomes shorter with decreasing /, so that in the dirty limit: £(/) = /, G < & ) (2.11) and in the clean limit: £(/) = 6 , (l > 6 ) (2.12) Eq. (2.11) and Eq. (2.12) are valid only for T = OK. In the clean limit (/ > £0) at T = OK, the magnetic penetration depth is the London penetration depth given in Eq. (2.1). The actual coherence length £(T) and the observed penetration depth A(T) as determined by microscopic theory are given in the clean limit (I ^> £0) as [21]: £(T) = 0.74£o ( j r ^ y ) 2 (2-13) A(T) = 0 . 7 1 A L ( — ^ ) 2 (2.14) Chapter 2. Theory 12 and in the dirty limit (/ <C £o): £(T) = 0.85(60" (T^TY (2-15) A(T) = 0.62AL (^j 2 ( jT3y) 2 (2.16) where Equations (2.13) through (2.16) are valid only in the neighborhood of Tc, such that (Tc — T)/Tc <C 1 [22]. The important result is that according to Eqs.(2.15) and (2.16), as I decreases {i.e. the superconductor becomes more impure), A(T) increases, while £(T) decreases. Thus A(T) ^> £(T) at all temperatures in an impure material. The high-temperature superconductors have short coherence lengths on the order of £ ~ 1 2 t o l 5 A . Since the electron mean free path / is typically ~ 150A in these materials, then they are well within the clean limit [23]. 2.1.6 B C S T h e o r y for Convent ional Superconductors In 1957 the underlying microscopic theory of superconductivity in metals was unveiled by J. Bardeen, L.N. Cooper and J.R. SchriefFer [24], in the now famous BCS theory. In normal metals, the situation is well described by free electron theory, where the elec-trons behave as free particles and the metallic ions play a limited role in conductivity. BCS theory outlines how in the presence of an attractive interaction between electrons (Cooper pairs), the normal state of an otherwise free electron gas becomes unstable to the formation of a coherent many-body ground state. The mechanism behind the weak attractive force binding the Cooper pairs was actually first suggested by Herbert Frolich [25]. He proposed that the same mechanism responsible for much of the electrical re-sistivity in metals (i.e. the interaction of conduction electrons with lattice vibrations) leads to a state of superconductivity. This hypothesis of an electron-phonon interaction was born out of experiments which found that the critical temperature Tc varied with Chapter 2. Theory 13 isotopic mass. In simple terms, an electron interacts with the lattice by virtue of the Coulomb attraction it feels for the metallic ions. The result is a deformation of the lattice (i.e. a phonon). A second electron in the vicinity of the deformed lattice corre-spondingly lowers its energy, resulting in an electron-electron attraction via a phonon. Viewed in this context, the superconducting order parameter \&(r) from GL-theory can be interpreted as a one-particle wave function describing the position of the center of mass of a Cooper pair [26]. Despite being an extremely weak attraction, bound pairs form in part because of the presence of a Fermi sea of additional electrons. As a result of the Pauli exclusion principle, electrons that would prefer to be in a state of lower kinetic energy cannot populate these states because they are already occupied by other electrons. Thus a Fermi sea is required to ensure the formation of bound pairs of electrons; otherwise an isolated pair of electrons would just repel one another as a result of the Coulomb force between them. The Fermi sea itself is comprised of other distinct bound pairs of electrons. It follows that each electron is a member of both a Cooper pair and of the Fermi sea which is necessary for the formation of all Cooper pairs. The force of attraction between the electrons which comprise a Cooper pair has a range equivalent to the coherence length. It should be noted that the separation between electrons in a Cooper pair (and thus the correlation length £), for a type-I superconductor, is large enough that millions of other pairs have their centers of mass positioned between them. It is then assumed that the occupancy of a bound pair is instantaneous and uncorrelated with the occupancy of other bound pairs at an instant in time [27]. Armed with knowledge of the fundamental particles responsible for superconductivity (i.e. Cooper pairs), the substitutions q = 2e and | ^ 0 | 2 = ns/2 immediately transform the Ginzburg-Landau result for the penetration depth X(T) [i.e. Eq. (2.6)] into the result predicted by London theory [i.e. Eq. (2.1)]. Chapter 2. Theory 14 One of the most remarkable features emerging from BCS theory, is the existence of an energy gap A(T) between the BCS ground state and the first excited state. It is the minimum energy required to create a single-electron (hole) excitation from the superconducting ground state. Thus the binding energy of a Cooper pair is two times the energy gap A(T) . BCS theory estimates the zero-temperature energy gap A(0) as [26]: A(0) = 1.76kBTc (2.17) and near the critical temperature Tc, A(T) / T \ 1 / 2 — ~ = 1.74 1 - — , T^TC (2.18) A(0) V TJ ' so that the energy gap approaches zero continuously as T —» Tc. Superconductors which obey Eq. (2.17) are considered to be weakly-coupled, in reference to the weak interaction energy between electrons in a Cooper pair. Furthermore, the wave functions corresponding to electron pairs (Cooper pairs), are spatially symmetric like an atomic ^-orbital with angular momentum L = 0. That is to say, the wave function of a pair is unchanged if the positions of the electrons are exchanged. This immediately implies that the spin part of the wave function is antisymmetric in accordance with the Pauli exclusion principle. In particular, the electron pairs are in a spin-singlet state 5 = 0 with antiparallel spins. The pairing mechanism in a conventional superconductor is thus appropriately called, s-wave spin-singlet. The energy gap of an s-wave superconductor is finite over the entire Fermi surface. Under ideal circumstances, the magnitude of the gap is the same at all points on the Fermi surface. BCS theory assumes the Fermi surface is spherical (see Fig. 2.3). More realistically however, the energy gap reflects the symmetry of the crystal under consideration [28]. For a conventional s-wave Chapter 2. Theory 15 superconductor with a weak-coupling ratio A ( 0 ) / & B T C = 1.76, BCS theory predicts: ^ 2 - 1 S = 3 . 3 3 ( | ) , / 2 e - ^ ( 2 , 9 ) for small T (i.e. T < 0.5TC) [17,29]. At low temperatures, the energy gap is virtu-ally independent of temperature and much larger than the thermal energy &gT. The probability of exciting a single electron with energy Ek is then proportional to the Boltzmann factor e~Ek^hsT. The maximum value of this probability is proportional to e~A(°MkBT = e-1 7 6 Tc/T^ w m c ] 1 j s thg exponential factor appearing in Eq. (2.19). Therefore in conventional superconductors, X(T) shows an exponential decrease at low temperatures. If the value of the energy gap is not constant over the entire Fermi surface, then the minimum value of the gap determines the density of quasiparticle excitations at these low temperatures. Hence the topology of the energy gap is crucial in deciding the low-temperature behaviour of A(T). If the sample under investigation is riddled with impurities, then there will exist a broad range of transition temperatures ATC [30]. Equations (2.17) and (2.18) suggest that one should anticipate a corresponding distribution of gap energies in such materials. Some systems (e.g. lead, mercury) produce experimental results which deviate sub-stantially from the BCS results [26]. These materials are more appropriately described by strong-coupling theory where the coupling ratio A(0)/ksT c is greater than the BCS prediction of 1.76. Under certain conditions superconductivity can occur without an energy gap in some materials. Tunneling experiments on superconductors with specific concentrations of paramagnetic impurities show this to be possible [31]. Theories exist which explain such anomalies, and the nature of the gap as we will soon see is a vital property to be considered in any theory describing superconductivity in the high-Tc compounds. Chapter 2. Theory 16 (a) (b) Figure 2.3: Isotropic s-wave energy gap for a (a) spherical Fermi surface and, (b) a cylindrical Fermi surface. The dotted lines outline the Fermi surfaces. Chapter 2. Theory 17 2.1.7 T y p e - I I S u p e r c o n d u c t o r s and the Vortex State As presented in their original form, the above results pertain almost exclusively to type-I superconductors. The main topic of interest here however, is the magnetic properties of type-II superconductors, whose magnetization curve is depicted in Fig. 2.4. Below Tc, in the presence of an externally applied magnetic field H, three distinct phases are recognizable, dependent on the strength of the applied field. Below a lower critical field izci(T) the superconductor is in the Meissner state with full expulsion of magnetic flux from its interior. For an applied field above an upper critical field HC2(T), magnetic flux fully penetrates the type-II material and returns it to its normal state. Within Ginzburg-Landau theory, the lower and upper critical fields may be written as: Hcl(T) and, Hc2(T) respectively. If the applied field lies between i J c l (T) and HC2(T), there is a partial penetration of flux into the sample leading to regions in the interior which are superconducting and others which are in the normal state; this is often referred to as the mixed state. Ginzburg and Landau defined type-II superconductors as those with K > l / v 2 . The high-temperature superconductors are extreme type-II, with large GL parameters K (i.e. A ^> £), large upper critical fields HC2(T), and small lower critcal fields HC\(T) [32,33,34]. In 1957, the Russian physicist Alexei A. Abrikosov predicted the existence of type-II superconductors by considering the solution of the GL equations for K > l / v 2 [35]. In particular, he considered the case where the externally applied magnetic field H is only (2.20) 4TTA(T)2 2TT£(T) 2 (2.21) Chapter 2. Theory 18 - M A M.Hci M.HC / i0Hc2 •< »—« *—* »-Full Meissner Vortex State Normal State Figure 2.4: Magnetization curve for a type-II superconductor. slightly below HC2(T); for in this region one obtains approximate solutions resembling those of the linearized GL equations [13]. The solutions revealed the presence of a periodic microscopic magnetic field distribution, transverse to the applied field. More precisely his efforts predicted a periodic square array of thin filaments of magnetic flux in the mixed state. Consequently, literature sometimes refers to the mixed state as the Abrikosov regime or for reasons to soon be made apparent, the vortex state. In the core region of a filament, the magnetic field is high and the material does not superconduct here; that is these regions are in the normal state. The magnetic field is screened from the rest of the sample by supercurrents which circulate around each filament. It is common to refer to the filaments as vortex lines and the array of filaments as a vortex lattice. At the vortex center where the magnitude of the local field is largest, the density of superconducting electrons ns and hence the order parameter \&(r) is zero. As one moves radially out from the center of a vortex core, ns increases and the supercurrents screen a greater amount of flux. At a radius on the order of the coherence length £(T), ns approaches the value in the bulk of the sample. It follows that the density of the supercurrents is greatest near the edge of the vortex core where the screening of the local magnetic field is at a maximum. Outside of the Chapter 2. Theory 19 vortex core, magnetic field leaks into the superconducting regions of the sample ( see Fig. 2.5), in much the same way magnetic field penetrates the surface of the semi-infinite superconductor in Figure 1.2. The superconducting order parameter is, for all intents and purposes, constant beyond regions of the order £(T) around the vortex cores [36]. It may be noted here that Abrikosov's prediction of the vortex state was remarkable in that it precluded any concrete experimenital proof of its existence. He derived the following relations between Hc\, Hc2, and the thermodynamic critical field Hc: Hcl = — (In K- 0.27) (2.22) K and, Hc2 = V2KHC (2.23) At a field Hc2 the vortex cores begin to overlap and there is no longer any solution of the Ginzburg-Landau equation (i.e. the material returns to its normal state) [37]. Hc\ marks the magnetic field at which it first becomes energetically favourable to have a flux line penetrate the superonductor. In the dirty limit (/ <C £0), one can write the upper critcal field as [14]: Hc2 = ^{XL/l)Hc = ^ _ (i) (,24) Thus it is clear that increasing the impurity content of the superconductor and thus shortening the mean free path I results in a higher value of Hc2 for the material. One may wonder how London's magnetic penetration depth A could be incorporated into a theory for type-II superconductors. Although London's original theory is indeed valid for A ^> £, it makes no mention of an upper critical field Hc2{T). Nevertheless, one could anticipate some extension of the London model applicable to type-II super-conductors by consideration of specific features. To start with, F . London theoretically Chapter 2. Theory Vortex Core 20 Figure 2.5: Magnetic flux leaking from the vortex cores where the field is highest into the surrounding superconducting regions (shaded area). The relative magnitude of the local magnetic field is depicted by the solid line. predicted that magnetic flux is quantized in a superconducting annulus. An analogous treatment of the supercurrents that shield the vortex cores, leads one to deduce in several different ways [13,17], that the flux associated with each vortex is quantized and defined by: he $ 0 = — = 2.07 x 10_ 7G-cm2 (2.25) where <3>0 is called the flux quantum, h is Planck's constant and c is the speed of light. London's original calculation of the flux quantum $ 0 was off by a factor of 2. The reason for this was his unknown knowledge of Cooper pairs. Experiments done on superconducting rings verify Eq. (2.25) as the correct equation [15]. For an isolated vortex line, the field as determined in the extreme type-II limit K > 1 is [35]: SM = ibK° (i) (2-29) where r is the distance from the axis of the vortex line, and K0 denotes a modified Bessel function of the second kind. For small distances (£ < r <C A), from the vortex-line axis, Chapter 2. Theory 21 Eq. (2.26) reduces to [13]: while further away (r ^> A): ^ = 2^ln(7) ^ B(r) = 7?h\hre-r/X (2-28) v ; 2TTA2 V 2r v ' The interaction between vortex lines becomes significant when their separation is < A [38]. In a type-II superconductor, the local magnetic field outside a vortex core decays in an exponential manner so that one is justified in describing the field in this region by the London equations. In a crude sense, the mixed state of a type-II superconductor may be visualized as a type-I superconductor with a regular array of normal state regions (i.e. the vortex cores) embedded in its interior. The magnetic penetration depth A is then a measure of the length over which magnetic flux leaks into the superconducting regions that encompass the vortex cores. The existence of the mixed state can be argued through free energy considerations. Consider a sample in an applied magnetic field Happued with a normal phase and a superconducting phase separated by a single interface. Realistically, the changeover between the two phases is not abrupt . Magnetic field leaks into the superconducting region a distance A(T), resulting in a contribution to the free surface energy of the interface that is negative. In addition, the order parameter vE'(r) decreases to zero over a distance £(T), thereby decreasing the actual volume of the sample which supercon-ducts. This leads to a positive-energy contribution at the interface. The sum of both phenomena give the net surface energy per unit length at the interface as [39]: 1 2 ' " = ^KvUeAeiT) - A2(T)] (2.29) A type-I superconductor where £(T) > A(T), has positive surface-energy; while type-II Chapter 2. Theory 22 superconductors with A(T) > £(T), have negative surface-energy. A more precise calcu-lation in the context of GL-theory predicts that the sign of the surface energy changes at K = A(T)/£(T) = I / A / 2 [17]. For a type-I superconductor in an applied field, the presence of normal-superconducting interfaces would increase the total free energy den-sity because of positive surface-energy contributions. This situation is of course most unfavourable. For this reason, in the intermediate state of a type-I superconductor large normal regions will be formed which contain many flux quanta [13]. On the other hand, in the case of a type-II superconductor, negative surface-energy contributions from normal-superconducting interfaces would reduce the free energy density. Conse-quently, the total energy is minimized by introducing as many of these interfaces as possible. This is exactly what happens in the mixed state of a type-II superconductor. It has been argued that the vortex model borne out of Abrikosov's solution of the Ginzburg-Landau equations is the most energetically favourable form for the magnetic flux to assume inside the superconductor [37]. Maximizing the surface-to-volume ratio of the normal regions leads to the lowest energy situation in the mixed state, because this provides the smallest positive surface-energy contribution. To achieve this, one can form normal regions which constitute either lamina of very small thickness (~ £) or alternatively, filaments of small diameter (~ £). Theoretical calculations indicate that the latter formation is lowest in energy for A ^> £ [14]. Since Abrikosov's original prediction of a square vortex lattice, subsequent solu-tions of the GL equations for magnetic fields just below the upper critical field HC2(T), have convincingly shown that a periodic triangular array of vortex lines has the lowest free energy of all possible periodic solutions, and hence is the most stable configura-tion [13,40,41]. Decoration experiments which utilize small iron particles to make the vortices visible through an electron microscope have not only provided the best con-firmation of the triangular arrangement but have also verified the flux quantum of Eq. Chapter 2. Theory 23 (2.25) [42]. It is clear from such experiments which verify the flux quantum that it is energetically more favourable to increase the number of vortices as one increases the magnetic field, rather than increasing the amount of flux in each vortex [37]. Thus in an isotropic type-II superconductor, the vortex lines repel one another as a result of the magnetic force between them, to form an equilateral triangular lattice in accordance with energy minimization. Chapter 2. Theory 24 2.2 T h e Pair ing M e c h a n i s m The mechanism responsible for superconductivity in the copper oxides is still a matter of serious debate. This section provides a qualitative comparison between the s-wave pairing state associated with conventional metallic superconductors and a proposed J-wave pairing mechanism for the high-temperature superconductors. The latter has attracted considerable attention in the past few years because it is experimentally verifiable. However, conflicting experimental results have left the nature of the pairing mechanism still very much a mystery. This chapter will avoid the rigorous mathematical details associated with each mechanism; the interested reader is referred to references within. 2 .2 .1 .s-Wave Pairing As touched upon earlier, conventional BCS superconductors are characterized by a s tandard s-wave, spin-singlet pairing state with S = 0, L = 0 Cooper pairs. The two electrons of a pair have equal and opposite momenta k and —k, so that the centre-of-mass momentum of a Cooper pair is zero. For an attractive electron-electron interac-tion, the bound state is symmetric upon exchange of electron positions, so it must be an antisymmetric singlet upon exchange of electron spins to satisfy the Pauli exclusion principle. Thus, at any instant of time, one can think of the electrons in a Cooper pair as being in a state (&,- f, — &; J,), and the wavefunction describing the pair consists of all states "z" occupied by the pair during its lifetime. An attractive interaction between two electrons results in a potential energy contribution which is negative, and thus low-ers the total energy of the electron system. The negative potential energy associated with a Cooper pair is the binding energy of that pair. Chapter 2. Theory 25 In normal metallic superconductors, the attractive interaction between electrons originates from the electron-phonon interaction, which is short range and retarded in space-time [43]. The lattice deformation resulting from the first electron takes a finite time to relax, and thus a second electron can be influenced by the lattice deformation at a later time. The attraction arising from the electron-phonon interaction overcomes the screened Coulomb repulsion between electrons due to the presence of other electrons and ions in the solid [26]. This screening aids in reducing the natural Coulomb repulsion between two electrons, leading to an effective interaction which is relatively short range compared with the unscreened Coulomb potential [38]. The net effect of the attractive interaction on all the other electrons in the material renders the normal Fermi liquid state unstable. Consider the normal ground state of a metal in the absence of an attractive electron-electron interaction at T = OK. The kinetic energy (and hence the total energy) of the system is minimized by requiring that the momenta of the plane wave states of the electrons fill up a sphere of radius PF = %]ZF (i.e. a Fermi sphere) in three-dimensional momentum space. Fig. 2.6(a) illustrates a Fermi sphere of radius kp in three-dimensional A:-space for a free electron gas at T = OK. The corresponding normal density of states N(E), where N(E)dE is defined as the number of electron states with energy between E and E -f- dE, is shown in Fig. 2.6(b). N(0) denotes the density of states at the Fermi surface at absolute zero. Original BCS theory described superconductivity in metals, so that the Fermi sur-face was assumed spherical in the normal state. In the BCS ground state (i.e. the state of the superconductor at T = OK), some of the electron states just outside the normal Fermi surface are occupied, and some just inside the Fermi surface are unoccupied. Certainly such an arrangement has a higher kinetic energy than does the normal state of the metal at T — OK. However, the BCS ground state is in part comprised of Cooper Chapter 2. Theory 26 (a) 5> K (b) Figure 2.6: A metal at T — OK. (a) The corresponding filled Fermi sphere, (b) The density of states. Chapter 2. Theory 27 pairs, the formation of which lowers the potential energy of the system. In the BCS ground state the arrangement of Cooper pairs is such that lowering of the potential energy outweighs the increase in kinetic energy, so that the BCS ground state has a lower total energy than that of the normal ground state [44]. As mentioned, the decrease in potential energy is due to the phonon-induced, electron-electron attractive interaction. The phonon-induced attractive force scatters Cooper pairs from one state (k | , —k j ) to another state (k' | , —k' J,). An electron oc-cupying state (k | ) near the Fermi surface vibrates the lattice resulting in the emission of a phonon of wave vector q. The electron is thereby scattered to a state (k' | ) , where k' = k — q. A second electron occupying state (—k J.), absorbs the phonon thus scat-—* —* —* tering to a state (—k' J,), where —k' = —k + q. The total centre-of-mass momentum in the final state is thus K = k'' + (—k') = 0, which is unaltered from before the scattering process. The two electrons forming a Cooper pair are continually scattered between states with equal and opposite momentum. Since only the total centre-of-mass momen-tum is conserved in such scattering processes, the momenta of the individual electrons is continually changing, so that one cannot explicitly assign a single momentum to each electron in the pair. Each scattering process reduces the potential energy of the electron system further. The negative potential energy contribution is greatest for scattering between states of equal and opposite momentum [39]. The range of momenta available to the scat-tered electrons in the ground state is dictated by the energy of the phonon and the Pauli exclusion principle. The phonon-induced attractive interaction can only affect those electrons in the vicinity of the Fermi surface. Electrons further inside the Fermi sphere cannot scatter to other levels because of the Pauli exclusion principle. To scat-ter electrons well below the Fermi level into unoccupied states would require phonon Chapter 2. Theory 28 frequencies much larger than that generated from the electron interaction with the lat-tice. Electrons closer to the Fermi surface may form Cooper pairs; however, as they do so the number of states available to the scattering electrons decreases. As the number of scattering events decreases, so does the maximum amount by which the potential energy of the system can be lowered. The formation of Cooper pairs must then cease at the moment the increase in kinetic energy due to moving electrons above the Fermi level exceeds the amount by which the potential energy is lowered. This limit then specifies the arrangement of the BCS ground state. One of the most distinctive consequences of BCS theory is that an energy gap opens up between the ground state and the lowest excited state. As shown in Fig. 2.7(a), the energy gap A(k) has the same symmetry as the Fermi surface of the normal state. The important feature in s-wave pairing is that the wave vector dependence of the energy gap is finite everywhere on the Fermi surface at temperatures below Tc. As the temperature is increased above T = OK, thermally excited phonons become available to scatter the electron pairs. Because of the energy gap in the excitation spectrum, excitations cannot occur with an arbitrarily small amount of energy as in the case of a normal metal. Phonons with energy comparable to twice the energy gap (2A) will scatter electrons in a Cooper pair to states above the gap. The electrons from the pair will no longer have equal and opposite momentum, so that their interac-tion potential becomes negligible and the Cooper pair is destroyed. Of course at low temperatures, the density of phonons with this much energy is small. Near Tc on the other hand, phonons with energy on the order of the energy gap are plentiful, and pair breaking is greatly enhanced. In addition, the gap itself is temperature dependent and can be well approximated by [15]: ~TCA(T)~ A(T) A(0) = t a n h T A(0)_ (2.30) Chapter 2. Theory 29 Figure 2.7: A BCS superconductor at T = OK. (a) The Fermi surface (dotted line) for the groundstate. (b) The corresponding density of states. The shaded region represents the occupied states of the superconducting electrons. Chapter 2. Theory 30 except near Tc where Eq. (2.18) better describes the behaviour. As T increases, the amplitude and frequency of the lattice-ion movement increases. This disrupts the propagation of the phonons between correlated electron pairs, resulting in a weakening of the attractive interaction and a corresponding reduction in the size of the energy gap [45]. Fig. 2.7(b) and Fig. 2.8 show the associated density of quasiparticle states N(E) at T = OK and T slightly above absolute zero, respectively. The states which are no longer occupied between Ep — A and EF+A pile up on the edges of the gap region. Above the energy gap, the electrons are often referred to as normal electrons or quasiparticles. —* —* That is, if an electron occupies a state (k f) then the state ( — k J.) need not be occupied. The corresponding density of quasiparticles (i.e. the normal fluid density) is usually denoted nn. Below the energy gap the electrons are paired. These electrons are referred to as superconducting electrons, and their associated density (i.e. the superfluid density) is denoted ns. At T = OK all the electrons are superconducting, while for T >TC they are all normal. For temperatures in between, the system is a mixture of superconducting and normal electrons. 2.2.2 T h e Mot ivat ion for an Al ternat ive Pairing M e c h a n i s m For the high-Tc superconductors, there are several experimental and theoretical reasons for seriously questioning traditional BCS theory with a simple phonon-induced electron-electron interaction. Experimentally, an extremely small isotope effect measured for YBa2Cu,307_<5 is often cited as one such deterrent [17]. BCS theory predicts an isotope shift if Tc is determined by the motion of the oxygen ions; however, substitution of 1 S 0 for 1 6 0 in YBa2Cu307_^ does not significantly change Tc. Ruling out a phonon-induced pairing mechanism based on these observations is premature however. The presence of an isotope shift implies that the lattice is certainly involved in the pairing Chapter 2. Theory 31 N(E) A N(0->-Thermally excited k\\\\pf- quasiparticles .^ \\\\w;. ^ E EF-A Ef EF+A Figure 2.8: The density of states for a BCS superconductor at T > OK. As T is increased above absolute zero, Cooper pairs are broken and the resulting normal electrons occupy states above Ep + A. mechanism. However, one cannot assume that the lack of an isotope effect necessarily implies that the pairing mechanism does not involve phonons. Superconductors such as ruthenium and zirconium exhibit virtually no isotope effect, while uranium shows a negative isotope effect [46]. The problem is that there are many additional factors which can effect the strength of the isotope shift. Thus, by itself the lack of a significant isotope effect in YBa2Cu307_,5 is not enough to rule out an electron-phonon mechanism. The anomalous normal-state properties of the cuprates suggest that these materi-als are not just a normal Fermi liquid above Tc, and therefore may not be adequately described by BCS theory below Tc. The electrical dc resistivity p(T), exhibits a linear dependence in temperature over a wide range of temperatures above Tc. For a conven-tional Fermi liquid associated with normal metals, p(T) ~ T2. This is a manifestation of the long lifetime of electrons near the Fermi surface in a conventional Fermi liquid [47]. The nuclear spin-lattice relaxation rate T{~l(T) shows a temperature dependence Chapter 2. Theory 32 substantially different from that of normal metals. Other anomalous normal-state prop-erties of the copper-oxide superconductors include the thermal conductivity «(T), the optical conductivity cr(cu), the Raman scattering intensity S(u>), the tunneling conduc-tance as a function of voltage g(V), and the Hall coefficient Rn(T) [48]. All of these normal-state properties are quite uncharacteristic of the Fermi liqtiid usually associ-ated with the normal state of conventional superconductors. In fact, it is possible that the unusual normal-state properties of the high-Tc compounds cannot be appropriately described by a Fermi liquid. Some argue on theoretical grounds that BCS theory cannot explain the high transi-tion temperatures of the cuprates. For example, it has been suggested that an electron-phonon mechanism probably cannot account for transition temperatures in excess of 40 to 50K [49]. The magnitude of the electron-phonon interaction required to generate a Tc comparable to that of YB^CnsOr^s, would substantially weaken the lattice. This structural instability would greatly reduce the density of electron states at the Fermi surface and hence destroy superconductivity [18]. This argument is not accepted by all. It has been suggested that the calculations leading to the above conclusion are not valid, so that an electron-phonon mechanism may still be the basis for superconductivity in the high-temperature superconductors [50]. 2.2.3 d-Wave Pairing It is plausible that the notion of Cooper pairing and BCS theory may still be applica-ble to the high-temperature superconductors, yet the nature of the pairing mechanism may be something other than the phonon-induced electron-electron interaction. The formation of a bound state can be achieved by any attractive interaction capable of overcoming the natural Coulomb repulsion between two electrons. Several alternative sources for this attractive force which are compatable with conventional BCS theory Chapter 2. Theory 33 have been proposed. One such mechanism, which has received much attention in re-cent years, is an electron-electron interaction mediated by magnetic spin fluctuations [51,52,53]. The concept is not entirely new. A similar process is believed to help facil-itate p-wave spin-triplet pairing (L = 1, S = 1) in superfiuid 3He, and to lead to other pairing states in certain organic superconductors and heavy fermion systems such as UPt 3 [49,53]. The antiferromagnetic state of the parent materials such as YBa 2 Cu 3 06 and the anomalous normal-state properties of the high-Tc superconductors provide the inspi-ration for a t tempts at describing the superconducting properties in terms of a spin-fluctuation exchange mechanism [52,53,82]. A logical starting point for such a theory is to suggest that the physical origin of those normal-state features which differ from nor-mal metals may somehow be responsible for superconductivity in the cuprates. It has been suggested that the measured anomalous normal-state properties of YBa2Cu307_5 stem from strong antiferromagnetic correlations of spins, and these same antiferromag-netic spin fluctuations are also responsible for superconductivity in the cuprates [1]. NMR measurements of the normal state have been successfully modelled with a nearly antiferromagnetic Fermi liquid [55,56,57]. Some insight into the origin of possible antiferromagnetic spin fluctuations in the superconducting phase of YBa2Cu307_§ may be obtained by examination of the an-tiferromagnetic insulating compound YBa 2 Cu 3 06. The structures of YBa 2 Cu30 6 and YBa 2 Cu 3 07 appear in Fig. 2.9 and the phase diagram for YBa 2Cu 30i r (6 < x < 7) appears in Fig. 2.10. In reference to Fig. 2.9(a), the Cu( l ) or C u - 0 chain layer of YBa 2 Cu 3 06 consists entirely of Cu 1 + ions. The singly ionized Cu ions have no mag-netic moment. Oxygen doping places O ions along the &-axis, resulting in a progressive conversion of Cu 1 + into Cu 2 + with the development of holes in the 3c?-shell of the Cu ions [59]. Chapter 2. Theory 34 (a) (b) ° Copper Figure 2.9: (a) The structure of the insulator YBa2Cu306 and (b) the structure of the superconductor YBa2Cu3C>7 [18]. Chapter 2. Theory 35 Oxygen Formula Content "x " Figure 2.10: Phase diagram for Y l ^ C u s O a ; , as a function of oxygen formula concen-tration x. Also shown is the phase diagram for a Zn-doped sample [58]. The Cu(2) or Cu02 planes of YBa 2 Cu 3 06 have predominantly Cu 2 + ions. Each Cu gives up two electrons; an electron from the 45-shell and the other from the 3<i-shell. The absence of an electron in the 3d-shell (a hole) results in a net magnetic moment (spin) on the Cu ions in this layer. Oxygen cannot easily be removed or added to the Cu02 planes. The oxygen concentration can be varied appreciably only in the C u - 0 chains. As mentioned, adding oxygen converts the copper ions in the C u - 0 chains from Cu 1 + to Cu 2 + . Beyond x £=: 6.5 it is believed that adding oxygen is equivalent to adding holes to the Cu02 planes. The oxygen which is randomly added to the chains becomes O 2 - by trapping two electrons which are believed to originate from the creation of two holes in the oxygens of an adjacent Cu02 plane. However, Hall coefficient measurements suggest that holes may also be forming in the chains [23]. Neutron difffraction and muon precession experiments indicate that the Cu mo-ments are antiferromagnetically aligned in YBa2Cu306 below the Neel temperature TN Chapter 2. Theory 36 (see Fig. 2.10). The Cu 2 + spins (i.e. spin 1/2 holes) in the Cu02 planes are coupled antiferromagnetically through a superexchange process with the oxygen ions. The O 2 -ions themselves have no net magnetic moment. At sufficiently low temperatures, the Cu ions in the chains also become antiferromagnetically ordered and couple with the Cu2 + ions in adjacent CuC>2 planes [18]. In view of this, the Cu 1 + labelling of the chain layer Cu ions may not be entirely accurate. Neutron and Raman scattering experiments sug-gest that the exchange interactions within the Cu02 planes are much greater than the coupling between adjacent layers. This is likely due to the greater separation between Cu ions in the vertical direction and a lack of O 2 - ions between adjacent Cu02 planes. The difference between strengths of the interplanar and intraplanar couplings means that Y B a 2 C u 3 0 6 exhibits a quasi two-dimensional magnetic behaviour. Furthermore, the measured maximum magnetic moment on the Cu ions is substantially smaller than what one would expect for a localized Cu 2 + ion. This may be due to the enhanced thermal fluctuations associated with a two-dimensionally ordered system [18]. It is a widely accepted belief that the electrons responsible for conduction in the copper-oxide superconductors are more or less confined to the Cu02 planes [52]. If this is the case then it is plausible that these electrons are paired by way of a two-dimensional system of antiferromagnetic spin fluctuations. The next obvious question to ask is, "Do these spin fluctuations persist in the superconducting phase?" Raman and neutron scattering measurements suggest that the magnetic fluctua-tions do indeed survive into the superconducting state. The spin-correlation length is substantially diminished in the superconducting state, but the amplitude of the mag-netic moments is not greatly diminished [18]. Furthermore, NMR data taken above Tc indicate the presence of two-dimensional antiferromagnetic spin fluctuations arising from the nearly localized Cu 2 + rf-orbitals in the Cu02 planes. Chapter 2. Theory 37 It is difficult to give an intuitive description of pairing due to spin-fluctuation ex-change. It is clear that a single hole will help destroy the antiferromagnetic order. However it is less clear whether it will at tract or repel a second hole with the same or opposite spin. The answers to these questions seem to depend very much on the regions of A;-space and r-space considered [60]. Weak-coupling calculations of the normal and superconducting state properties have been carried out [1,57,82] for an antiferromagnetic spin-fluctuation induced interaction between quasiparticles on a two-dimensional square lattice. Such a two-dimensional model is at best an approximation to the behaviour of the three-dimensional YBa2Cu307_5 compound. Nevertheless, these calculations yield a value of the transition temperature Tc which is near 90K and a superconducting pairing state with dx2_y2 symmetry. For this pairing state the energy gap is of the form: A(k, T) = A 0 (T) | cos(kxa) - cos(kya) | (2.31) where A 0 (T) is the maximum value of the energy gap at temperature T and a is the lattice constant or distance between nearest neighbor Cu atoms in the plane. The angular momentum and spin of a Cooper pair is L = 2 and S = 0, respectively (i.e. singlet dx2_y2 pairing). The superconducting gap originating from the spin-fluctuation mediated interaction has a momentum (or k) dependence, in contrast to the phonon-frequency dependence of the gap associated with an electron-phonon interaction. It is clear from Eq. (2.31) that the excitation gap vanishes when \kx\ = \ky\. Fig. 2.11 shows the four nodes which result along the diagonals in the Brillouin zone at the Fermi surface. In three-dimensional A>space the gap vanishes along four nodal lines running parallel to the Ar.-axis for a cylindrical Fermi surface, or along four nodal lines joining the north and south poles for a spherical Fermi surface (see Fig. 2.12). Because of these nodal lines, Chapter 2. Theory 38 kx—ky >k Figure 2.11: The the energy gap for dx2_yi symmetry. The gap is largest along the kx and ky-directions. Nodes exist at the Fermi surface (dashed circle) along \kx\ = \ky\. there will be considerably more quasiparticle excitations at low temperatures compared to conventional s-wave superconductors. Thus even at T = OK there is a quasiparticle contribution to the supercurrent [61]. For YBa2Cu307_5 which is highly anisotropic, a cylindrical Fermi surface with no gap in the kz direction seems like a plausible de-scription. However, it has been suggested that the cross-section of the Fermi surface in YBa2Cu307_i is not a perfect circle. If this is the case, then the pairing state can have c?-wave symmetry but not give rise to nodes in the gap [49]. To gain a qualitative understanding of the symmetry in Fig. 2.11, consider the two-dimensional square lattice of antiferromagnetically ordered spins depicted in Fig. 2.13. Chapter 2. Theory 39 (a) 3» k. (b) s-k. Figure 2.12: c?x2_y2-symmetry for a (a) spherical Fermi surface and (b) a cylindrical Fermi surface. The Fermi surfaces are shown as dashed lines. Chapter 2. Theory 40 a K X a A Figure 2.13: A square lattice of antiferromagnetically arranged spins. In real space nearest-neighbor spins are separated by the lattice constant a. Such an ar-rangement is a simplified model of the Cu02 planes in YBa2Cu3C>7. In the YBa2Cu307 compound, a « b, so that the Cu02 planes are almost square and the localized spins of Fig. 2.13 correspond to the antiferromagnetically-correlated spin fluctuations asso-ciated with the Cu 2 + e?-orbitals. In the derivation of Eq. (2.31), a spin-spin correlation function (electronic spin susceptibility) was chosen which gave a good quantitative fit to NMR measurements of the Knight shift and the spin-lattice relaxation rates of 6 3Cu, 1 7 0 and 89Y nuclei in YBa 2 Cu 3 07 [82]. The electronic spin susceptibility is represen-tative of the strength of the spin-fluctuation-mediated pairing potential. This function is strongly peaked at the nesting wave vector Q — ( ± £ , ± - ) in the first Brillouin zone. For an s-wave gap, the electronic spin susceptibility is suppressed at ( ± - , ± - ) [2]. Chapter 2. Theory 41 To understand the significance of Q, consider first the phase-space restrictions on the electron-electron scattering ra te for a conventional electron gas assuming a cylindrical Fermi surface. At T = OK the Fermi cylinder is full and there are no electrons to scatter from such that energy and momentum are conserved. For T > OK, an excited electron with energy different than Ep can scatter into a shell of partially occupied levels —* centered about Ep in &-space. That is to say, the range of momenta available to the scattered electron is proportional to the temperature T [47]. This situation is depicted in two dimensions in Fig. 2.14(a), where the incoming quasiparticle momenta are k and — k. Any interaction between electrons changes the momenta of the quasiparticles such that k' & k±q. Any orientation of the wave vector q in Fig. 2.14(a) will yield the same available phase space for electron scattering near the Fermi surface at a temperature T. Consider now a nearly antiferromagnetic Fermi liquid with nesting vector Q as shown in Fig. 2.14(b). In this nested-Fermi liquid, two electrons with momenta (k, —k) near the Fermi surface exchange the antiferromagnetic spin fluctuation which has a —* sharp peak at Q. The two electrons are subsequently scattered (by the oscillating potential set up by the corresponding spin density wave) to states with wave vectors (k', —k') near opposite sides of the Fermi surface. In this case k' « k±Q. As illustrated in Fig. 2.14(b), the range of momenta available to the scattering electron is greater than —* —* —* in the conventional Fermi liquid of Fig. 2.14(a). For all k — k' = Q parallel to the wave vectors ( ± - , 0 ) and ( 0 , ± - ) , the available phase space for scattering is the same. As k — k' = Q is rotated away from these directions, the range of momenta available to the —* scattering electron decreases so that it is smallest when Q = ( ± f , ± f )• The evolution —* —* of the wave vector k — k' in Fig. 2.14(b) maps out the gap function of Fig. 2.11. This is demonstrated in Fig. 2.15. As k — k' is rotated away from ( ± - , 0 ) or (0, ± £ ) , the Chapter 2. Theory (a) >K (b) A jrsSSSSS^r--.(Si) Figure 2.14: Phase space available for electrons scattered from the state (k, —k) to the state (k',—k') at temperature T. (a) An electron gas system with a weak attractive interaction, (b) A nested Fermi liquid. The dashed square represents the first Brillouin zone. In both cases the Fermi surface is indicated by a solid line. The shaded regions indicate the available phase space which is proportional to T. Chapter 2. Theory 43 range of momenta available to scatter into decreases and so does the magnitude of the energy gap. It should be noted that there are theories which predict dx2_y2-wsive pairing which are not based on spin fluctuations [28]. Considering these, it seems appropriate to dis-cuss dx2_y2 symmetry as it pertains to the Fermi-surface geometry, rather than to intro-duce the details of the theoretical calculations which predict dx2_y2-wa,ve pairing from antiferromagnetic spin fluctuations. The orientation of the Fermi surface in Fig. 2.14(b) and Fig. 2.15 is that used to explain commensurate peaks at ( ± - , ± - ) in neutron ex-periments involving Y I ^ C u s C v [62]. The corners of the Fermi surface are actually more rounded than they appear in these figures. The Fermi surface for La2_a;Sra;Cu04 has more curvature in the sides and is rotated 45° from that of YBa2Cu306.95 so that incommensurate peaks are observed at ( ± - , ± - ) in neutron experiments. Returning to YBa2Cus06.95 and the notion of spin fluctuations, if one assumes that a given oxygen nucleus is coupled predominantly to the spins on its two nearest-neighbor Cu sites, then the Cu spin-density of states is greatest near the Brillouin zone corners ( ± £ , ± - ) [56]. Also since the 0 nucleus is resting between two oppositely directed Cu spins, the transferred hyperfine field from the Cu moments cancels at the 0 site so that the spin density vanishes there. Thus the Cu spins relax the 0 nuclei so that the dominant contribution to the spin susceptibility comes from the Cu 2 + d-orbital spin states. Thus low temperature excitations may result from the influence of the Cu spins on the superconducting carriers. Opponents of the spin-fluctuation mechanism have argued that the measured quasi-particle lifetimes in YBa^CusOr are much too short for the quasiparticles to take advan-tage of this sort of interaction [53]. Strong-coupling calculations (which normally imply Chapter 2. Theory (a) (b) Figure 2.15: The dependence of the available phase space on the scattering wave vector k — k'. As the phase space decreases, the energy gap increases so that it is at its maximum in (b). The dashed square represents the first Brillouin zone and the Fermi surface is indicated by a solid line. Chapter 2. Theory 45 a short quasiparticle lifetime) have been carried out. Results show that one still ob-tains a Tc of 90K for YBa2Cu307, with a dx2_y2 pair gap [53]. Subsequent calculations of the anomalous normal-state quasiparticle properties in the corresponding strong-coupling regime have also been done [53,63]. Results are consistent with experimental measurements, indicating that the strong-coupling calculations are reasonable. Theoretically a d-wave pairing state is appealing because it avoids the strong on-site Coulomb repulsion which is inherent in an s-wave pairing state. Also a spin-fluctuation pairing mechanism for the high-Tc superconductors would not lead to a lattice instabil-ity as may occur for an extremely strong electron-phonon interaction. Unfortunately it is yet to be shown if the dx2_y2 symmetry evolves out of calculations for oxygen concentrations less than in YBa2Cu307. Also a complete microscopic theory for dx2_y2 pairing is still unavailable. On the experimental front, a t tempts at determining the pairing state in the copper-oxide superconductors have been conflicting and inconclu-sive. Chapter 2. Theory 46 2.3 Ani so tropy in YBa 2 Cu30 7 _ f t 2.3.1 General Cons iderat ions The electronic properties of high-Tc superconductors are extremely anisotropic. Con-sequently theories which describe their behaviour must be able to distinguish between the motion of the superconducting carriers in different directions. For YBa2Cu307_s, the crystal structure is orthorhombic (see Fig. 2.9). The layered crystal structure of YBa2Cu307_j results in a strong a-c anisotropy, while the orthorhombic distortions due to the C u - 0 chains introduce another a-b anisotropy. The anisotropic nature of various physical quantities measured in single crystals of YBa2Cu307_,s have been well docu-mented. For instance, the electrical resistivity along the a and c-directions of the unit cell varies linearly with temperature, while along the 6-direction there is a distinct up-ward curvature for increasing temperature, which has been at tr ibuted to conductivity along the C u - 0 chains [64,65]. Critical-field, critical-current and magnetic penetration depth measurements in YBa2Cua07_5 also exhibit strong anisotropy. The anisotropy of the magnetic penetration depth A in the uniaxial high-Tc materials is a direct con-sequence of the superconducting currents not being isotropic [66]. In addition to the penetration depth exhibiting a large a-c anisotropy, recent measurements on untwinned YBa2Cu306.95 single crystals show a small but significant a-b anisotropy [10]. 2.3.2 A n i s o t r o p y of the Magne t i c P e n e t r a t i o n D e p t h In experiments to determine A, single crystals are much preferred over poly crystalline samples. This is true for several reasons. Because YBa2Cu307_5 is strongly anisotropic, the orientation of the sample in the applied field is significant. With single crystals one has control over this feature. By positioning the single crystal with its c-axis parallel Chapter 2. Theory 47 to the applied field, one can readily proceed to determine the magnetic penetration depth in the a6-plane (or C u - 0 plane), Aa{, (or A||). In an analogous fashion, suitable orientation of the single crystal in the applied field in principle allows measurement of Xc or A_L (i.e. the magnetic penetration depth perpendicular to the C u - 0 planes). This is much more difficult to perform experimentally, however, since the crystals grow in such a way that the a-b dimensions are much greater than in the c-direction. With polycrystalline samples, the principal axis of each grain is randomly oriented with respect to the applied field. Consequently one must average over all possible orientations of the c-axis to simulate the field distribution and then try to extract a value for the magnetic penetration depths \ab and Ac. This could be difficult, since different combinations of values of Aa(, and Ac may give similar line shapes. Further considerations attached to the use of powdered samples include the dis-similarity in shape of the individual grains. Consequently, each grain has a different demagnetization factor and thus a slightly different average field. This leads to an ad-ditional broadening of the field distribution, which if not properly taken into account will lead to an underestimate of the magnetic penetration depth. In general, the magnetic penetration depth in an anisotropic superconductor is de-termined by replacing the effective mass m* of the superconducting electrons by an effective-mass tensor m* [67]. Until very recently, the a-b anisotropy had been consid-ered negligible, so it has always been assumed that , for the uniaxial high-temperature superconductors, m* has a degenerate eigenvalue m*b (i.e. m* « m£ = rn*ab) associated with supercurrents flowing in the afr-planes that screen magnetic fields perpendicular to the planes, and a nondegenerate eigenvalue m* associated with supercurrents flowing along the c-axis, which help screen magnetic fields parallel to the C u - 0 planes [68]. Chapter 2. Theory 48 Thus for a uniaxial, anisotropic superconductor: m*LC2 Kb = xhr^T- (2-32) an> d m* c2 V 47re2n<. ' One can define an anisotropic ratio 7 for uniaxial superconductors, such that: Ac /m* ( A £ 2 (2.34) Aa6 V "»:* \| ( A 5 2 ) i For Y B a 2 C u 3 0 7 , 7 ^ 5 [69]. 2.3.3 Ani so tropy of the Energy G a p In any superconductor with a non-spherical Fermi surface, one expects anisotropy in —* the superconducting gap function A(k). In general, the gap function will vary according to the angle with respect to the crystalline axes [70]. That is, the energy required to —* break a Cooper pair will depend on the direction of k. In YBa2Cu307, one expects the size of the energy gap to show some variation in the aft-plane. However this does not necessarily imply nodes in the gap. If the Fermi surface in the aft-plane is not a perfect circle, then the gap will certainly be anisotropic; but as is the case for an anisotropic s-wave pairing state, the gap may remain finite over the entire Fermi surface. Furthermore, if we are to think of the energy gap in the superconducting state of YBa2Cu307_«5 as having dx2_y2 symmetry, a-b anisotropy will produce nodes in the gap which are not precisely along \kx\ = \ky\. As mentioned above, conventional superconductors also have some anisotropy in the gap function. However, in most of these materials the mean free path is such that / <C £ and also A <C £, so that the anisotropy is negligible when interpreting the experimental Chapter 2. Theory 49 results [71]. In the high-Tc superconductors where ( < I < A, the anisotropy in the gap might play a significant role. Chapter 3 Measur ing t h e pene tra t ion d e p t h w i th TF- / /+SR 3.1 T h e Field Dis tr ibut ion The vortex lattice in a type-II superconductor results in a spatially varying magnetic field B(r) inside the superconductor. Transverse -field muon spin rotation (TF-^i+SR) can be utilized to investigate the local magnetic field distribution function [72,77]: ,T,/N J 6[B'- B(r)]d3r t , The function n(B) denotes the field density. Specifically it is the probability that the magnitude of the magnetic field at a point f inside the superconductor is B. Eq. (3.1) defines n(B) as the spatial average of a one-dimensional Dirac delta function. By definition /+oo n(B)dB = l (3.2) -oo The field distribution pertaining to a perfect vortex lattice free of distortions resembles that of Fig. 3.1. The magnetic lineshape exhibits van Hove singularities corresponding to: A: the minimum value of B(r) which occurs at the center of the vortex lattice. B: the saddle point value of B(r) which occurs midway between two vortices. C: the maximum value of B(r) which occurs at the vortex cores. 50 Chapter 3. Measuring the penetration depth with TF-/J,+SR 51 a) / \ \ \ / / / I I 1 [(X)) ) I ^ . , < / \ v < \ \ \ \ A X / — ^ x \ \ \ ^ OVH^-tt 'O) \ \ \ < - ^ / / / -" / L \ ^ . ^ / J I I ~7 1 (b) B n(B) c \ B Figure 3.1: (a) A single unit cell in the vortex lattice. The dotted lines represent contours of the local field around a vortex core. The local field is perpendicular to the page, (b) The corresponding field distribution. Chapter 3. Measuring the penetration depth with TF-/J,+SR 52 The average local magnetic field strength inside the superconductor is denoted B, The spatial average of B is just the first moment of n(B) [73]: r+oo B /+oo n(B) B dB = fi0H + (1 - N)M (3.3) -oo where H is the magnitude of the externally applied field, N is the demagnetization factor (0 < N < 1) and M is the magnetization of the sample. The demagnetization factor N depends on the shape of the sample. The second moment of n(B) determines the width of the field distribution and is given by: n(B) (B - B) dB (3.4) -oo If the applied field H ~ -ffC2/4 then the London model is valid. At such fields the spacing L between adjacent vortices is large compared to the coherence length. In the London theory, the local magnetic field in an isotropic superconductor is given as [68]: Bs / R eift-? where the sum extends over the reciprocal lattice vectors K of the vortex lattice. Combining (3.4) and (3.5), the second moment of n(B) for a triangular flux lattice as determined in the London picture is [32,68]: (AB2) = 0.00371 $^ A-4 (3.6) It is clear from Eq. (3.6) that one can estimate the penetration depth A from the second moment ( A S 2 ) , so that A is inherently related to the line width of the measured field distribution. Eq. (3.6) is valid at intermediate fields 2iJc l < H < Hc2/4: where the second moment (AB2) and hence A is independent of the applied field [34,74]. At such fields the inter-vortex spacing L <C A [66]. Rammer argues that in the case of YBa2Cu307_A-, Eq. (3.6) is useful only for fields significantly smaller than 50kG (i.e. Chapter 3. Measuring the penetration depth with TF-/J,+SR 53 5.0T), and above this significant deviations from the London model result rammer. For low fields Hcl < H <C Hc2, where L• ^> A and the vortices are well separated, the second moment as determined in the London limit is related to the average local field B approximately by [74]: {AB2) = B2 [ 2V5TT ( A V / 3 / 2 L ) 2 1 + 4TT2 ( V / 3 A / 2 L ) 2 } (3.7) This result was derived for a triangular flux-line lattice. According to Eq. (3.7), increasing the applied field in this region leads to a corresponding increase in the second moment. This statement of course assumes that the penetration depth A is a field independent quantity. At high fields the second moment decreases with increasing applied field H as the vortex cores start to overlap. The second moment at such fields can be estimated by using Abrikosov's numerical solution of the GL-equations for H < HC2'. (AB2)^CAe0X-4[l-b)2 (3.8) where b = B/Hc2 and 7.52 x 1 0 - 4 < CA < 8.19 x 10 - 4 , with the lower limit given by [34] and the upper limit given by [74]. London theory is not valid in this high-field regime, where the inter-vortex spacing of the lattice is comparable to £ [i.e. L < (5 — 10)£] and the vortex cores start to overlap [75]. In London theory, where £ is assumed to be zero, the local magnetic field in the center of a vortex is not finite. To get around this divergence, Brandt [34,72] approxi-mates the vortex core by a gaussian function of width £, which provides a dome-shaped peak for the field. An additional term (1 — b) can be incorporated to roughly account for the field dependence of the superconducting order parameter [76]. As in Eq. (3.8) this term is significant for applied fields close to the upper critical field iJc 2 , leading to a noticeable reduction in the width of the magnetic lineshape. Including this effect, Chapter 3. Measuring the penetration depth with TF-/J,+SR 54 the vortex core can be approximated by a gaussian function of width £ / \ / l — b. Thus, according to Brandt [34,72,76], one can modify Eq. (3.5) so that the local magnetic field of an isotropic superconductor for applied fields H < Hc2/4: is m = BJ: — — - ^ — (3.9) R l + i-b The second exponential in Eq. (3.9) introduces an upper cutoff for the reciprocal lattice vectors at K ss 27r/£ [77]. This value emerged from Brandt 's numerical solution of the GL-equations [34,72]. The upper cutoff for K leads to a finite value of the local field B at the vortex core. Fig. 3.2 illustrates the field distribution B — B as determined by Eq. (3.9) for the average internal magnetic fields B=5kG and 15kG. As the field is increased, the distance between vortex cores decreases. Also the difference between the maximum and minimum field in the distribution decreases with increasing applied field. Fig. 3.3 shows the effects on the field distribution of increasing the magnetic penetration depth A, while Fig. 3.4 illustrates the dependence on the GL-parameter K. The variation of the supercurrent density Js(r) with position r from the vortex cen-tre can be determined by substitution of Eq. (3.9) into Eq. (2.3). Fig. 3.5 illustrates the dependence of the absolute value of the supercurrent density distribution on magnetic field. The supercurrent density is zero at the centre of a vortex core, corresponding to the maximum in the field distribution; Js rises steeply to its maximum value at the edge of the vortex core and then falls off exponentially. The singularity observed halfway between two vortices denotes a change in the flow direction of the supercurrents. Fig. 3.8 shows the dependence of the average value of the supercurrent density Js on the average field B for YBa2Cu307_,>,. As the applied field ( « B) is increased above the lower critical field, the average supercurrent density rises steeply. This is because the supercurrents which screen the vortex cores have less room to spread out at higher fields when the cores are closer together. However, as shown in Fig. 3.8, the relationship Chapter 3. Measuring the penetration depth with TF-/i+SR 55 is not linear. This is due to the overlapping of vortices, which tends to reduce Js. As the field is increased, the inter-vortex separation decreases. Eventually the overlapping of vortices dominates the contribution to the average supercurrent density and the curve turns over, so that Js decreases with further increase in the applied magnetic field. Interestingly, the turnover in Fig. 3.8 occurs around 50kG; the maximum field at which Rammer claims Eq. (3.6) is valid [32]. The dependence of the supercurrent distribution on A and K can be seen in Fig. 3.6 and Fig. 3.7, respectively. As mentioned in the previous chapter, the orientation of the applied field with respect to the crystallographic axes of the superconductor is important . For an applied field parallel to the c-axis, the second moment of n(B) valid at intermediate fields can be written as [66] (AB\ = 0.00371 $1 \-J (3.10) and for an applied field parallel to the a6-plane, (AB2}± = 0.00371 $1 (Aa6Ac)~2 (3.11) In the latter case, the supercurrents flow across the afr-plane and along the c-axis. In fact, for the uniaxial anisotropic high-Tc superconductors, there exists a nonzero component of magnetic field transverse to the vortex axes, when the applied field is not directed along one of the principal axes in the afe-plane or along the c-axis [78]. As the angle between the applied field and the crystallographic c-axis is increased, the vortex lattice distorts. The circular cross-section of an isolated vortex line associated with an applied field directed parallel to the c-axis (see Fig. 3.1) is replaced by an elliptical cross-section as shown in Fig. 3.9. This is a result of the anisotropy of the supercurrents. Correspondingly, the equilateral-triangular vortex lattice is stretched into an isosceles one. The peak in the measured ^ + S R field distribution n(B) will be split because now there are two types of saddle points. However, because the vortices Chapter 3. Measuring the penetration depth with TF-/j,+SR 56 r (lira) r (Mm) .00 r (/xm) r (/xm) Figure 3.2: The field distribution between the centers of two vortices for (a) £?=5kG and, (b) B=15kG. The field distribution between the center of a vortice and the center of the flux lattice for (c) B=5kG and (d) B=15kG. In all cases A = 1490A and K = 68. Chapter 3. Measuring the penetration depth with TF-/J+SR 57 r (/xm) 0 . 3 0 -0 .25-0.20-S 0.15-m 0.10-00 0.05 -0.00--0 .05-~ - ^ \ v 1 1 1 1 (d) -C^— i i .000 .005 r (jim) .010 .015 r M .020 .025 Figure 3.3: The field distribution between the centers of two vortices for (a) B=5kG and (b) I?=15kG and the field distribution between the center of a vortice and the center of the flux lattice for: (c) B=5kG and (d) 5 = 1 5 k G , where A = 1490A (solid line) and A = 2236A (dashed line). In all 68. Chapter 3. Measuring the penetration depth with TF-/J,+SR 58 .03 .04 r M 0 . 3 0 -0 .25-0.20-J< 0.15-m 0.10-1 m 0.05 -0.00-- 0 . 0 5 -O^ O (b) V^JJ. ^ r .00 .01 .02 r (um) .03 .04 020 .025 r Gu.m) r ( H Figure 3.4: The field distribution between the centers of two vortices for (a) B=5kG and (b) i?=15kG and, the field distribution between the center of a vortice and the center of the flux lattice for (c) I?=5kG and (d) i?=15kG, where K — 68 (solid line) and K = 30 (dashed line). In all cases A = 1490A. Chapter 3. Measuring the penetration depth with TF-fj,+SR 59 .03 .04 r (urn) .00 1.2-1.0-^ 0 . 8 -i <"0.6-^ ' 0.4 -0.2-0.0-Ao _ i (c) ---.01 .02 r (fj.m) .03 .04 0.0 .000 r (fim) Figure 3.5: The absolute value of the supercurrent distribution between the centers of two vortices for (a) B=5kG and (b) i?=15kG. The supercurrent distribution between the center of a vortice and the center of the flux lattice for (c) B=5kG and (d) B~15kG. In all cases A = 1490A. and K = 68. Chapter 3. Measuring the penetration depth with TF-/J,+ SR 60 Figure 3.6: The absolute value of the supercurrent distribution between the centers of two vortices for (a) B=5kG and (b) I?=15kG and the supercurrent distribution between the center of a vortice and the center of the flux lattice for (c) i?=5kG and (d) 5 = 1 5 k G , where A = 1490A (solid line) and A = 2236A (dashed line). In all cases K = 68. Chapter 3. Measuring the penetration depth with TF-/J,+ SR 61 Figure 3.7: The absolute value of the supercurrent distribution between the centers of two vortices for (a) B=5kG and (b) 5 = 1 5 k G and, the supercurrent distribution between the center of a vortice and the center of the flux lattice for (c) i?=5kG and (d) Z?=15kG, where K = 68 (solid line) and K — 30 (dashed line). In all cases A = 1490A. Chapter 3. Measuring the penetration depth with TF-/J,+SR 62 200 B (kG) Figure 3.8: The variation of the average supercurrent density Js on the average mag-netic field B in YBa 2 Cu 3 0 7 -5 with A = 1490A and « = 68. Chapter 3. Measuring the penetration depth with TF-/i+SR 63 | I I • ' l I • ' . A A ' ' • V 2X X2 1 1 • i • i ' • - 1 - 1 - 4 - - ^ ^ J - - n ' - J -i . i i 1 Figure 3.9: The vortex lattice associated with an applied field nonparallel to the crys-tallographic c-axis. The applied field is perpendicular to the page and the dashed lines are contours of the local magnetic field around the vortex cores; 1 and 2 indicate the two different saddle points in such an arrangement. Chapter 3. Measuring the penetration depth with TF-/J,+SR 64 themselves are stretched, the saddle points are comparable; so that experimentally, a broadening of the peak in n(B) may be observed, rather than a splitting. The width of the field distribution n(B), as evidenced by Eqs. (3.10) and (3.11), is of course influenced by the anisotropic distortions of the vortex lattice. However it is clear from Eq. (3.10), that by orientating the sample with its crystallographic c-axis parallel to the applied field, the problem becomes isotropic in nature. Of course this is only true if one is ignoring the smaller a-b anisotropy. 3.2 T h e Role of the Pos i t ive M u o n The elegance of the ^ + S R technique is the employment of a positive muon //+ to mi-croscopically probe the local magnetic field distribution of Eq. (3.1) in the bulk of the superconducting sample. Thus, measurements of the penetration depth using / /+SR are less sensitive to the sample's surface than other conventional techniques which measure A. A positive muon has the following properties [77,79]: Rest mass: m^ = 105 .65839(29)%^ = 206.7291(ll)m e = 0.11261(28)mp Charge: +e Spin: S^ = \ Magnetic Moment: \i^ = 4.84 x 10~3fiB = 3.18//p Average Lifetime: r^ = 2.19709(5)//s Muon Gyromagnetic Ratio: 7^/2TT = 0.01355342(±0.51ppm) ^ p Muons are generated via the weak decay of pions: 7T+ —> /I+ + i/M (3.12) A low energy / i+-beam can be created by making use of the fact that some of the Chapter 3. Measuring the penetration depth with TF-/j,+SR 65 pions which are produced in collisions of high energy protons with a fixed carbon or beryllium production target stop inside that target. These pions rapidly decay (7^ ?« 26ns) according to Eq. (3.12). Muons produced in this manner are often referred to as surface muons to distinguish them from higher momentum muons which are obtained from pions which decay while in flight. A //+ emerging from 7r+-decay has the direction of its spin antiparallel to its momentum vector. This result follows from conservation of linear and angular momentum, and the fact that the neutrino uM is a left-handed particle (i.e. the neutrino helicity = —1). A beamline composed of magnetic dipole and quadrupole magnets is used to collect muons emitted in a given direction. Consequently, these muons are nearly 100% spin polarized, with their spin pointing backwards relative to their momentum. In the TF-/U+SR arrangement (see Fig. 3.10), the yu+ spin is rotated perpendicular to the muon momentum using a Wien filter or separator, which consists of mutually perpendicular electric and magnetic fields, both of which are transverse to the beam direction. After passing through a collimator, the spin-polarized positive muons are implanted one by one into the sample, which is situated in an external field H. The applied field is transverse to the initial muon spin polarization, and hence parallel to the muon momentum. The muons entering the sample rapidly thermalize (stop) within ~ 10~13s. Their low momentum implies a short range in the sample: R ~ 120mg/cm , which corresponds to a thickness of approximately 200/Um in YBa2Cu306.95 [77]. Inside the sample, the muon's magnetic moment comes under the influence of the —* local magnetic field. For YBa2Cu306.95, the local magnetic field Hiocai is the vector sum of the externally applied field Hext, the magnetic field produced by the circulating —* —* supercurrents Hs, and the field H^ resulting from local magnetic moments: Hiocai = Hext + Hs + H^ (3.13) Chapter 3. Measuring the penetration depth with TF-/j,+SR 66 L spin A l\ fi —beam R < -H applied Figure 3.10: Schematic diagram for standard transverse-field geometry. The incoming muon spin is perpendicular to the muon momentum. The muons pass through a thin muon counter M and stop in the sample 5", where their spin precesses around the transverse local field. Decay positrons are detected in the L and R counters. The muon spin rotates somewhat in the applied field before reaching the sample. Chapter 3. Measuring the penetration depth with TF-/j,+SR 67 The muons precess about the local field Hiocai with a Larmor frequency: <^V = ln{HoHlocal) = -j pB local (3.14) where Biocai is the local magnetic induction and 7^ is the muon gyromagnetic ratio. If the muons occupy identical sites in the sample, they will be subjected to the same local magnetic field. Therefore they will precess in phase with one another so that the magnitude of their initial polarization is retained. If, however, the muons experi-ence different magnetic fields in the sample, they will not precess at the same Larmor frequency. The resulting dephasing of their spins manifests itself in a corresponding loss of polarization. After a mean lifetime rM, the fj,+ decays into a positron and two neutrinos: / i + —> e+ + ue + uM (3.15) The distribution of decay positrons is asymmetric with respect to the spin polariza-tion vector P(t) of the muon, such that the highest probability of emission is along the direction of the muon spin. Consequently, the time evolution of the muon spin polarization P(t) can be monitored, since the fx+ reveals its spin direction at the time of decay. A thin //+-detector placed in front of the sample registers the arrival of a muon in the sample and starts a clock. A pair of positron detectors (one on either side of the sample as in Fig. 3.10) can be used to detect the positrons emerging from the /J,+-decay. Once a decay positron is registered, the clock is stopped. For a single counter the number of positrons recorded per time bin At is N(t) = N°e-t,r- [1 + A°P(t)} + B° (3.16) where JV° is a normalization constant, rM is the muon lifetime, A0 is the maximum precession amplitude (or maximum experimental decay asymmetry), P(t) is the time Chapter 3. Measuring the penetration depth with TF-JJ,+SR 68 evolution of the muon spin polarization in the direction of the positron counter and B° is a time-independent background from unwanted events. 3.3 T h e R a w A s y m m e t r y For 2-Counter G e o m e t r y Consider the pair of counters i and j in Fig. 3.11, analogous to counters R and L respectively in Fig. 3.10. In a real experiment one can define the raw asymmetry Aij(t) of the appropriately paired histograms Ni(t) and Nj(t) defined by Eq. (3.16) as [77]: wh ere Aij{t) = [W) ~ B°] -[Ni(t) - B°] + N3\t) - B° Nj(t) Bi (3.17) so that Nr{t) = N°e-^ [1 + A° Pi(t)] + B° Nj(t) = i V J e - ^ [l + A) Pj(tj\ + B° Al3{t) = iV° [1 + A?Pi{t)] - N° N?[l + A?Pi(t)]+m l+A°P3{t) 1+A'Pjit) (3.18) (3.19) (3.20) The motivation behind introducing Aij(t) is the elimination of the muon lifetime (which is a well known quantity) and the random backgrounds B° and B°. In the ideal situation, the two counters in question are identical to one another so that the /U+SR histograms recorded by each differ only by an initial phase. Because the muons travel for some time in the applied field before reaching the sample, their initial direction of polarization at the time of arrival in the sample is field dependent. However, the difference in phase between histograms is due solely to the geometry of the positron counters with respect to the sample. This situation is illustrated in Fig. 3.11. In the Chapter 3. Measuring the penetration depth with TF-/J+SR 69 Figure 3.11: The orientation of the initial muon spin polarization P(0) with respect to a pair of counters i and j , in a TF- / / + SR setup. The applied field is out of the page and S denotes the sample. presence of a constant magnetic flux density B, the time evolution of the muon spin polarization vector P(t) is given in general by P(t) = COS(2TTUM + 9) (3.21; where v^ = ^ - is the muon spin precession frequency and 9 is the initial phase of the muon spin polarization vector. For counters i and j in Fig. 3.11, the polarization of the muon spin as seen by each counter is: P(t) = COS(2TTI/M + V>) Pj[t) = COS(2TT//W + 4) (3.22) (3.23) Since \ij)-<f>\= 180° then, Pi(t) = C0S[27TI/M + {4> + 7T)] = - C0S(27TI/M + 4>) = - P j ( t ) (3.24) Chapter 3. Measuring the penetration depth with TF-[i+SR 70 With the counters i and j arranged symmetrically along the x and — x directions: Pi(t) = Px(t) and, Pj(t) = -Pi(t) = -Px(t) (3.25) Thus Eq. (3.17) becomes: (N? - N°) + (N°A° + N°A°) Px(t) AtJ(t) = f-5 U- y i ^ 3—LL \1 (3.26) (N° + N°) + (N°A°-N°A°)Px(t) If N° = N° and A° = A° = A0 then Eq. (3.26) reduces to: Aij(t) = A°Px{t) (3.27) If B° = B°, then using Eqs. (3.17), (3.18), (3.19) and (3.27) the a:-component of the polarization can be written as: N? + m p*w - o, 'r*tlr» (3-28) 3 3.4 T h e Corrected A s y m m e t r y In practice, differences in counter efficiency and deviations from perfect geometry com-plicate the above treatment. If the sensitivity of the individual positron counters are not the same, the values of the normalization constant JV°, the maximum decay asym-metry A° and the random background levels B° will differ. The number of recorded events in each counter also depends on the solid angle they subtend at the sample. If the coverage of the solid angle is not maximized because of counter misalignments, there will be a decrease in the number of decay positrons detected. To account for these differences one can modify the above results by assuming: « = TT , P = TTA and> 7 = TT (3-29) N° ' H \A°\ ' ' B° v ' Chapter 3. Measuring the penetration depth with TF-fx+SR 71 Experimentally one can determine these constants by examining data above Tc where the muon precesses around a spatially constant field. With the above consid-erations, the raw asymmetry A{j(t) for the counters i and j can then replaced by a corrected asymmetry Acorrected(t). This is obtained by substituting Eq. (3.29) into Eq. (3.20) and defining A° = A0 and P{(t) = Px(t): A m - (l-«) + A°Px(t)(l + a[}) _ {l-a) + Araw{t)(l+a(3) AcorrecteAt) ~ ( 1 + a ) + Jop^Q ( 1 _ a 0 ) ~ ( 1 + a ) + ^ ( f ) ( l _ a f i ) ^-™) 3.5 T h e Re laxat ion Funct ion In the vortex state the muons experience a spatially varying field strength B(r). Con-sequently the ar-component of the fi+ polarization may be written: 1 N W) = ^  Ecos bMW + 8) (3-31) » = 1 where 6 is the initial phase. In a real superconductor there are additional contributions to the relaxation rate, so that a more appropriate description of the JJL+ polarization is: -, N Px{t) - Gx(t)- £ cos [j,B(fz)t + 8] (3.32) where Gx(t) is a relaxation function (or depolarization function), which describes the damping of the precession signal. The precise form of the relaxation function GS) depends on the origin of the depolarization. In many cases one assumes a gaussian form which is appropriate when the depolarization originates from the muon-nuclear-dipolar interaction [79]. There is no significant loss of polarization during the short time over which the muons thermalize. This is because the primary interactions by which the muons rapidly lose their initial kinetic energy are electrostatic in nature and hence do not affect the muon spin [80]. Loss of the muon spin polarization in the vortex state is primarily due Chapter 3. Measuring the penetration depth with TF-/J,+SR 72 to the inhomogeneous field distribution, which in turn can be related to the magnetic penetration depth A. As A decreases, the spatial variation in the magnetic field becomes greater and there is a corresponding increase in the relaxation rate of the muon spin polarization (see Fig. 3.12). As Fig. 3.1(b) indicates, the field distribution for a perfect vortex lattice is far from being gaussian, but rather is highly asymmetric. The interaction of the /x+-spin with nuclear-dipolar fields in the sample leads to further damping of the precession signal and a corresponding broadening of the field distribution. Normally, a gaussian distribution of the dipolar fields at the /x+-site is assumed. Above Tc, this leads to a Gaussian relaxation function: Gx(t) = e-<^2/2 (3.33) where G& is the muon spin depolarization rate due to the nuclear-dipolar fields. The value of ad determined from data taken above Tc, is assumed to be the same in the superconducting state. For a real sample in the vortex state, decoration experiments have shown that the vortex lattice is not perfect. Deviations from the ideal flux-line lattice lead to a further relaxation of the precession signal [72]. Consequently we can redefine the relaxation function as: Gx(t) = e-(*2+*?>2/2 = e-'W2 (3.34) where aj is the muon spin depolarization rate due to lattice disorder and any additional depolarizing phenomena and <re// is the effective depolarization rate such that a\^ = aj + cr2-. The influence of aejj is to further broaden the field distribution n(B) beyond that due to A. It should be noted that the additional broadening of the field distribution due to flux-line lattice disorder is difficult to define. Because the field distribution correspond-ing to an ideal flux-line lattice is highly asymmetric, one would anticipate distortions Chapter 3. Measuring the penetration depth with TF-/J,+SR 73 ce 0.3 0.25 0.2 " 0.15 e S o.i 2 0.05 -0 -0.05 (a) D a a i st** i i i i 2 3 4 TIME i t t i c r o s e c ) ce u. o 0.25 0.2 0.15 0.1 0.05 0 n nc "i 1 D a -s a a o - a a a _ 0 n i "•• - i - • r (b) -a a - 0 i i i i i 2 3 4 TIME ( f l ic rosec) Figure 3.12: Asymmetry spectra plotted in a rotating reference frame due primarily to the vortex lattice in single-crystal YBa2Cu306.95- The da ta is for an applied transverse field of 15kG at the temperatures (a) T=75K where A « 2040A and (b) T=2 .9K where A « 1526A. Chapter 3. Measuring the penetration depth with TF-fi+SR 74 of the lattice to also be asymmetric in nature. In Eq. (3.34) we are assuming the distortions are gaussian distributed, but it can be shown analytically that convoluting with a gaussian does not change the average field of the distribution. 3.6 Model l ing T h e A s y m m e t r y S p e c t r u m of t h e Vortex S ta te In terms of A, the experimental raw asymmetry for TF-//+SR analysis of the vortex state in an isotropic type-II superconductor can be modelled assuming that the contribution to A(t) from a particular point in the flux lattice is e- ; t f-r- , e-A2 t f2 / [2«2( l-&)] ' A0 N •A-rawV1) = ~^T / ., COS f,BtJ2 1 + A"2 A2 + 0 (3.35) K - ' (1-6) where I have combined Eqs. (3.9), (3.27) and (3.31). Eq. (3.35) applies to an ideal flux lattice. To account for smearing of the field distribution due to flux-lattice disorder and nuclear dipolar fields, Eq. (3.35) must be convoluted with a gaussian distribution in the form of Eq. (3.34): •A-raw(t) N N cos t = i K -ii?-r%-A2A-2/[2«2(l-(>)] 1 + J-T2A2 (1-6) + 8 (3.36) Assuming that the flux-line lattice is composed of equilateral triangles as depicted in Fig. 3.1(a), an arbitrary reciprocal-lattice vector K — nk\ + mk-i can be written in terms of the real lattice vectors r\ = Lx and r2 — L/2(x + -v/3y) so that K = 2TT 1 /„ nx -)—y=(2m — n)y (3.37) w hevt L = I 2 $ 0 (3.38) is the distance between adjacent vortices. Combining Eqs. (3.36) and (3.37) gives Chapter 3. Measuring the penetration depth with TF-/.i+SR 75 A-awW A0 x cos < 7M** £ c o s { f [a,.n + a i ( ^ ) ] } e AW -A / 4-rr ^ ( n 2 + m 2 —mn «2 V v ^ / ^f7^ -cr 2 t2 TVj N2 i=o j=o x + (>&) ( ?i2 4-T7i2 — 771 n ) A2 (1-6) + 0 (3.39) where fij = a tx + ajj/ is a vector in real space and the sum over a, and a,- extends over a single unit cell. Eq. (3.39) can be substituted into Eq. (3.30) and the resulting theoretical corrected asymmetry used to fit the measured asymmetry. 3.7 4 -Counter G e o m e t r y and the C o m p l e x Polarizat ion In the 2-counter setup of Fig. 3.11, information is lost by failing to detect decay positrons emitted in the up and down directions. A more efficient setup is the 4-counter arrangement depicted in Fig. 3.13, where additional counters are placed on the y and — y axes. The L and R counters monitor the x-component of the polarization Px{t) as before, while the U and D counters measure the y-component of the polar-ization Py(t). Ignoring geometric misalignments and differences in counter efficiency, Py(t) differs from Px(t) only by a phase of 90°. In terms of the field distribution n(B), Px(t) and Py{t) are defined as: r+oo 2 2 / '+00 Px(t) = e'^ff1 / 2 / n(B) cos [llxBt + 6} dB J — oo (3.40) 2 1 i f~r°° Py[t) = e - ' W / 2 / n(B) cos [llxBt + 6- TT/2] dB J — oo 2 1 i / " " J - 0 0 = e - ^ / i / 2 / n(B) sin [j^Bt + 0} dB (3.41) where, Gx(t) = Gy(t) = e"""-^1 ' . For an applied field in the ^-direction, the muon spin polarization transverse to the magnetic, field Bz may be described by the complex Chapter 3. Measuring the penetration depth with TF-/J,+SR 76 L D Figure 3.13: The orientation of the initial muon spin polarization -P(O) with respect to a set of four counters L, R, U and D, in a TF-^+SR setup. The applied field is out of the page and S denotes the sample. quantity [81] P+(t) = PX(t)+iPy(t) — < T , , t2'2 I*™ n{Bzy^B^dBz J — oo (3.42) The real Fourier transform of the complex muon polarization P (t) approximates the field distribution n(Bz) with statistical noise due to finite counting rates [34]. At the later times in the asymmetry spectrum, most of the muons have already decayed. With few muons left, the statistics at these later times are low, resulting in increased noise at the end of the asymmetry spectrum. Using the fact that P (t) is defined only Chapter 3. Measuring the penetration depth with TF-fi+SR 77 for positive times t > 0, the field distribution n(Bz) may be written: n{Bz) ~ 2Re ( ^ [°° P + ^ e - ^ ^ e H ^ ' - ^ / / ^ 2 ) ^ } (3.43) 12ir Jo J where the gaussian apodization parameter a A is chosen to provide a compromise between statistical noise in the spectrum and additional broadening of the spectrum which such a procedure introduces. The complex asymmetry for the 4-counter setup is defined as: A+(t) = A°P+(t) = A°Px(t) + iA°Py(t) = Ax(t) + iAy(t) (3.44) where Ax(t) and Ay(t) are the real and imaginary parts of the complex asymmetry, respectively. The number of counts per second in the ith counter [i — R (right), L (left), U (up) or D (down)], may be written: Ni(t) = JV?e-'/T" [1 + Mt)} + B% (3.45) where Ai(t) = A°Pi(t) is the asymmetry function for the ith raw histogram. Rearranging Eq. (3.45): Ai{t) = e t/r„ Ni(t) - Bi - 1 (3.46) N° In terms of the individual counters depicted in Fig. 3.13, the real asymmetry Ax(t) and the imaginary asymmetry Ay{t) may be written: Ax(t) = \ [AR(t) - AL{t)} (3.47) M*) = \ lAu(t) - AD(t)] (3.48) In the present study, the real and imaginary parts of the asymmetry are fit simultane-ously. For the vortex state of YBa2Cu306.95, the real asymmetry can be fit with Eq. Chapter 3. Measuring the penetration depth with TF-/i+SR 78 (3.39). The imaginary part of the asymmetry can be fit with the same function, but with a phase difference of 90°. 3.8 T h e R o t a t i n g Reference Frame It is often convenient to fit the measured asymmetry in a rotating reference frame (RRF) . To do this, one subtracts a phase </>RRF = ^RRF^ from the phase of Eq. (3.39), where LORRF = 2-KVRRP is the chosen R R F frequency. The R R F frequency is taken to be slightly lower than the average Larmor-precession frequency u7M of the muon. The precession signal viewed in this rotating reference frame has only low frequency components on the order of <UM — OJRRF, where uJ^ is the average precession frequency in the lab frame. This has two important consequences. The first is that the quality of the fit can be visually inspected. Second and most important , it allows the da ta to be packed into much fewer bins, greatly enhancing the speed of fitting. Chapter 4 E x p e r i m e n t a l D e t e r m i n a t i o n of Xab(T) 4.1 S a m p l e Characteris t ics A mosaic of three twinned single crystals of YBa2Cus06.95 were utilized in this ex-periment. The YBa2Cu306.95 crystals were of extremely high quality, as characterized by low-field magnetization, afr-plane resistivity, microwave surface resistance and heat capacity measurements [85]. For example, the superconducting transition temperature Tc, determined by magnetization measurements was found to be 93.2K with a relatively narrow transition width ATC < 0.25K. The YBa2Cu.3O6.95 single crystals were grown by a flux method in yttria-stabilized zirconia crucibles. CuO-BaO flux was poured over hot (950-970°C) crystals initially grown from high purity Y 2 0 3 (99.999%), CuO (99.999%) and B a C 0 3 (99.997%) com-pounds. To ensure each crystal had the identical oxygen concentration corresponding to the maximum Tc, all three crystals were carefully annealed prior to the experiment. For 3 days the crystals were held in 1 atmosphere of O2 at 860°C, followed by 14 days at 450°C before rapidly cooling to room temperature. Together the three crystals had a total mass of 53mg. 4.2 T h e A p p a r a t u s The measurements were performed on the M l 5 surface /i+ beamline at TRIUMF. The beam consisted of low momentum (28.6MeV/c), 100% backward spin polarized positive 79 Chapter 4. Experimental Determination of Aa(,(T) 80 muons. The short range of these muons makes this beamline ideal for the study of thin samples. A crossed-field separator was used to rotate the muon spins perpendicular to their momentum direction. After passing through a 1cm diameter collimator and a thin muon defining counter, a small fraction of the incoming beam came to rest in the crystals (30 ,000/ i + s - 1 stops out of 200, 000/ / + s - 1 incoming). Great effort was devoted to reducing the background signal in this experiment, which superimposes itself on the measured field distribution originating from the sam-ple. The background signal originates from muons stopping in the sample holder or in the cryostat walls and windows, which then precess in the externally applied field. Previous / /+SR experiments on single crystals have been seriously plagued by the in-separable nature of the often large background signal [77]. A novel experimental setup was used to eliminate most of the background signal produced by those muons which missed the sample [86]. The three crystals were mounted on a thin layer of aluminized mylar using a minute portion of Apiezon N grease, with their c-axes parallel to the applied field. Recall with this orientation of the field it is Aa&(T) which is measured. The thin aluminized mylar provides no appreciable signal. The three crystals together provided a total area of 36mm2 for the incoming muon beam. Data taken with only one of the three single crystals of YBa2Cu306.95, showed no further appreciable reduction in the background signal and gave the same foreground signal as the 3-crystal sample. Consequently, a mosaic consisting of all three single crystals was used to maximize the counting rates for muons striking the sample. The mylar with the mosaic of crystals was stretched over the end of a hollow 4.45cm-diameter, cylindrical, aluminum sample holder, as indicated in Fig. 4.1. A horizontal 4He gas-flow cryostat with an internal diameter of 4.92cm, allowed cooling of the crys-talline YBa2Cu306.95 down to ~ 2.6K. A 7T warm-bore superconducting magnet called Chapter 4. Experimental Determination of Ao6(T) 81 0150mm WARM BORE OF 7T MAGNET VETO (V) COUNTER CRYOSTAT VACUUM SPACE LOCATION OF SAMPLE THERMOMETER He SPACE BEAM VACUUM COLLIMATOR SAMPLE MUON COUNTER MYLAR FOIL BACKWARD (B) POSITRON COUNTERS -SAMPLE HOLDER SAMPLE HOLDER TUBE RING Figure 4.1: The low background / / + SR apparatus . Chapter 4. Experimental Determination of \af,(T) 82 Helios was used to produce magnetic fields transverse to the initial muon spin direction. The positrons emitted from the muon decay were readily detected by a cylindrical arrangement of positron counters, coaxial with the magnet axis. Overlapping forward (F) and backward (B) counters were used to define a solid angle for muon decay events originating from the sample. A veto (V) counter in the form of a cylindrical scintillator cup, was employed to discriminate against those muons which missed the sample. A "good" muon stop was defined as M • V" (M = muon counter). A "good" positron stop (i.e. a positron originating from a muon that stopped in the sample) was defined as F-B-V. 4.3 T h e M e a s u r e d A s y m m e t r y Transverse-field / / + SR spectra with approximately 2 x 107 muon decay events were taken under conditions of field cooling in applied magnetic fields of 0.5T and 1.5T. Figure 4.2(a) and Fig. 4.2(b) show the real and imaginary asymmetry spectra pertaining to the YBa2Cu306.95 sample resting in an applied field of 0.5T and at a temperature well above Tc (i.e. T « 110K). For convenience the signals are displayed in a reference frame rotat ing at a frequency 2.3MHz below the Larmor precession frequency of a free muon. The average frequency of oscillation in Fig. 4.2(a) and Fig. 4.2(b) is determined by the applied magnetic field of 0.5T. Consequently, the corresponding frequency distribution [see Fig. 4.2(c)] exhibits a single peak at 67.3MHz related to the applied field through Eq. (3.14). Visual inspection of Fig. 4.2(a) and Fig. 4.2(b) suggests that the real and imaginary asymmetry spectra differ in phase by 90°, consistent with the discussion in the previous chapter. The solid curve passing through the data points is a fit to the data using a polarization function in the form of Eq. (3.21) with a relaxation function Chapter 4. Experimental Determination of \ab(T) 83 resembling Eq. (3.33), so that A(t) = A°P(t) = A°nst-a^t2l'2cos (27r*y + 6) (4.1) where A°ns (ns = normal state) is the precession amplitude, crns is the depolarization rate due to nuclear dipolar broadening and vM is the average precession frequency of a muon about the applied magnetic field. The real and imaginary parts of the measured asymmetry were fit simultaneously. The relaxation of the muon spin precession signal is small and is owing primarily to the distribution of the nuclear-dipolar fields in the sample. In particular, ans fa 0.13/is - 1 in Fig. 4.2. As one cools the sample below Tc, the relaxation rate of the muon precession signal increases due to the presence of the vortex lattice. The asymmetry spectra pertaining to a pair of counters for the YBa2Cu306.95 crystals in an applied field of 0.5T is shown for three different temperatures below Tc in Fig. 4.3. The signals are shown in a rotating reference frame 3.3MHz below the Larmor precession frequency of a free muon. As the muons stop randomly on the length scale of the flux lattice, the muon spin precession signal provides a random sampling of the internal field distribution in the vortex state. The ensuing asymmetry spectrum is a superposition of a signal resembling Fig. 3.12 originating from muons which stop in the sample and an inseparable background signal resembling Fig. 4.2(c), due to muons which miss the sample, do not trigger the veto counter and whose positron also does not trigger the veto counter. The origin of the residual background signal is still uncertain, but may be due to those muons which scat-ter at wide angles after passing through the muon counter and are thus not vetoed by the V counter. Fig. 3.12 was obtained by choosing a rotating reference frame frequency equal to the background frequency, determined by fitting the data. Unfortunately, the broadening of the background signal is not necessarily identical to the field distribution above Tc and thus cannot be fixed in the fits below Tc. Chapter 4. Experimental Determination of Xab{T) T I I I r 0 1 2 3 4 5 6 TIME (fis) LJ O r> tr _) 0. 2 < _J < LU Cd 0.2 0.17S 0.15 0.125 0.1 0.075 0. OS 0.025 -0 I ' I I • H I1 ' * 63 64 65 66 67 68 69 70 71 72 73 FREQUENCY (MHz) Figure 4.2: (a) The real part and, (b) the imaginary part of the muon precession signal in YBa2Cu306.95 at 110K in a magnetic field of 0.5T. The solid curves are a fit to the data assuming a gaussian distribution of fields [see Eq. (4.1)]. (c) The corresponding frequency distribution. The peak is at 67.31MHz, corresponding to an average field of 0.497T. Chapter 4. Experimental Determination of A0j(T) 85 (a) T=73.3K TIME Os) Figure 4.3: The muon precession signal for YBa2Cu306.95 in a field of 0.5T and at, (a) 73.3K, (b) 35.5K and (c) 5.8K. The solid curves are a fit to the data assuming the field distribution of Eq. (2.10). Chapter 4. Experimental Determination of Xab(T) 86 0.05 0.04 -Ld Q Q_ < _J en 0.02 -0.01 64 65 66 67 68 69 70 71 72 73 74 FREQUENCY (MHz) Figure 4.4: The corresponding Fourier transforms of Fig. 4.3 for YBa2Cu306.95 in a field of 0.5T and at, (a) 73.3K, (b) 35.5K and (c) 5.8K. Chapter 4. Experimental Determination of Xab(T) 87 The beat occurring in all three spectra of Fig. 4.3 is due to the difference in the average precession frequency of a muon in the internal field of the vortex lattice and the precession frequency of the muon in the background field. The solid curves in Fig. 4.3 are fits to the theoretical polarization function of Eq. (3.39). An additional polarization function in the form of Eq. (4.1) was added to model the background signal pertaining to the muons which missed the sample, so that the measured asymmetry is of the form: A(t) = Aaam(t) + Abkg(t) (4.2) where Asam{t) (sam = sample) is as given in Eq. (3.39) and the background asymmetry Abkg(t) (bkg = background) has the form: Abkg{t) = A ^ e ^ V 2 / 2 cos (^Bbkg t + d) (4.3) As the temperature is lowered, the vortices become better separated and the muon spin relaxes faster due to the presence of a broader distribution of internal magnetic fields. This is better displayed in Fig. 4.4 which shows the corresponding real Fourier transforms of Fig. 4.3. Recall from Eq. (3.6) that the width of the field distribution is proportional to l/X2ab(T). Thus it is clear from Fig. 4.4 that \ab(T) decreases with decreasing temperature. The sharp spike on the right side of each frequency distribution in Fig. 4.4 is at tr ibuted to the residual background signal. It has been determined to account for approximately 13% of the total signal amplitude at 0.5T and ~ 5% at 1.5T. Figure 4.5 shows the frequency distribution resulting from field cooling the sample in the 0.5T field [Fig. 4.5(a)] and then lowering the applied field by 11.3mT [Fig. 4.5(b)]. As shown the background signal shifts down by 1.5MHz and positions itself at the Larmor frequency corresponding to the new applied field. The signal originating from muons which stop in the sample does not appear to change under this small shift in field. This clearly demonstrates that at low temperatures the vortex lattice is strongly Chapter 4. Experimental Determination of Xab(T) 88 pinned. Furthermore, the absence of any background peak in the unshifted signal implies that the sample is free of any appreciable non-superconducting inclusions. The shifting of the background peak away from the sample signal has since been duplicated for higher temperatures and at other applied fields. Field shifts in excess of ~ 200G were not at tempted for fear that the crystals would shatter as a result of the strain exerted by the pinned vortex lattice. The asymmetric frequency distribution shown isolated in Fig. 4.5(b) has the basic features one would anticipate for a triangular vortex lattice, but with the van Hove singularities shown in Fig. 3.1(b) smeared out. Structural defects in the vortex lattice, variations in the average field due to demagnetization effects and a-b anisotropy are all possible reasons for a lack of sharper features. 4.4 D a t a Analys i s To determine the low temperature behaviour of A0(,(T) several assumptions were made in the fitting procedure to reduce the number of independent variables. To start with, the Ginzburg-Landau parameter K = \ab/t, was assumed to be independent of temper-ature. Although this is strictly valid only for weak coupling s-wave superconductors away from Tc, the lineshapes are not very sensitive to K in the low-field region being con-sidered here. Determining a value for K was accomplished by first fitting the recorded asymmetry spectra with s a s a fixed quantity. The value of K which minimized the sum of the x 2 ' s f ° r each temperature considered was then taken to be the best value for K. The value K = 68 gave the best overall fit to both the 0.5T and 1.5T data. Increasing K to 73 was found to change Aaf,(0) by less than 0.3nm. The value K = 68 is close to the value K = 69(1.4) determined from previous lineshape measurements on similar crystals in higher magnetic fields [77]. Chapter 4. Experimental Determination of Xab(T) 89 O X - i 1 r -(a) TF=0.5T (b) TF=0.489T . 64 66 68 70 72 FREQUENCY (MHz) 74 Figure 4.5: (a) The Fourier transform of Fig. 4.3(c) [i.e. the same as Fig. 4.4(c)] for YBa2Cu306.95 in a field of 0.5T at 5.8K. (b) Same as in (a) except that the field was lowered by 11.3mT after field cooling to 6K. Chapter 4. Experimental Determination of Xab(T) 90 Fits to the early part of the signal (i.e. the first 1/J.s) for data below Tc using an equation in the form of Eq. (4.1) with a single gaussian relaxation function of the form exp(—a2t2/2) and v^ pertaining to the average internal field, provides a simplified visual display of the dependence of the lineshape width on temperature. It is straightforward to use the polarization function in Eq. (4.1) to relate the relaxation parameter a to the second moment (AB2). The relationship is [68] ( A S 2 ) = -2 (4.4) % Comparing to Eqs. (3.6), (3.7) and (3.8), one has a oc -j- (4.5) Aab Because of the asymmetric shape of the true field distribution, using a gaussian distri-bution of internal fields gives poor fits to the data. However, the fits are sufficient to provide a crude estimate of the temperature dependence of the second moment. Fig. 4.6 shows the variation of the broadening parameter a for single gaussian fits of the 0.5T and 1.5T data. Both sets of da ta suggest that the width of the field distribution varies linearly with temperature below 20K. Furthermore, the broadening of the lineshape appears larger at these low temperatures for the applied field of 0.5T. The linear term appears to weaken slightly at 1.5T. Above 20K the data for the two fields are nearly identical. The single gaussian fits of course cannot determine \ab(T) explicitly, but are useful nonetheless in giving an approximate temperature dependence of Xab(T). They also help facilitate a comparison with previous studies where only single gaussian fits were possible. A more precise treatment of the data using the phenomenological model of Eq. (3.39), holds the two parameters <je// and l/\2b accountable for the width of the mea-sured field distribution in the sample. Clearly these two parameters must combine to Chapter 4. Experimental Determination of Aa(,(T) 91 0 20 40 60 80 TEMPERATURE (K) 100 120 Figure 4.6: The gaussian linewidth parameter a in YBa2Cu306.95, in magnetic fields 0.5T of (triangles) and 1.5T (squares). Chapter 4. Experimental Determination of Xab(T) 92 mimic the behaviour in Fig. 4.6. Since aeff and 1/A^6 both contribute to the linewidth and both are expected to be temperature dependent quantities, the two parameters cannot be treated as independent of one another when analyzing the data. Indeed, fits to the data in which both parameters were free to vary have crejf and 1/A^ playing off one another as in Fig. 4.7. A temperature point which appears locally high in the creyy vs. T plot, appears locally low in the 1/A^, vs. T plot and vice versa. A plot of l/A^, vs. creff suggests a linear correlation between the two parameters as shown in Fig. 4.8. The solid curve through the points in Fig. 4.8 has the following form: jr = ^ E ^ (4.a) Aab ^ where the relaxation parameter ans [see Eq. (4.1)] is determined by a run taken above Tc in the normal state and C is a constant {i.e. the slope) chosen so as to minimize the total x2 f ° r all runs in which T < 55K. In other words, only runs where T < 55K were considered in determining C since it is the low temperature regime which is of primary importance. Figure 4.9 shows the total \ 2 arising from global fits of the 0.5T and 1.5T data for various choices of the constant C. The proportionality constant C was found to be 0.0293(10)^mVs_ 1 and 0.0258(10) ium2/us_1 for the 0.5T and 1.5T data, respectively. The depolarization rate ans was approximately 0.13/us-1 and 0.11/zs-1 for the 0.5T and 1.5T fields, respectively. In the first type of analysis, the total asymmetry amplitude A° for signals recorded below Tc was fixed to the value of the precession amplitude A°ns obtained from fitting data above the transition temperature, prior to determining C in Eq. (4.6). Below Tc the asymmetry amplitude of the measured signal A° is the sum of the precession amplitude of the background signal (Alk ) and the precession amplitude of the signal originating from within the sample (A°am). Thus, here we are assuming that the total Chapter 4. Experimental Determination of \ab(T) en 3 . 1 . «t 1.35 1.3 1.25 1. 2 1.15 1 i — i -T • i e l l ! (a) < 'i i i • I I i I I ----i 10 15 20 25 TEMPERATURE (K) 30 CM I E =3. 51 . 5 50 47. 5 -45 CM I ^ 42. 5 -37.5 1 --r - I i 11 1 1 1 1 1 \ 1 - i I (b) t i i 10 15 20 25 TEMPERATURE (K) 30 Figure 4.7: The temperature dependence of (a) the gaussian linewidth parameter aejj and (b) the magnetic penetration depth Aaj, in the low temperature regime at a field of 0.5T, as deduced from fitting data with independent aeff and Xab-Chapter 4. Experimental Determination of Xab(T) 94 60 50 7 40 E O 30 20 10 -0 *• Aob-2=[geff2-(Q.i3)2]1/2 0.0293 0.25 0 .5 0 .75 a8ff (Mm-1) 1.25 1.5 Figure 4.8: The relationship between Aaj and the gaussian broadening parameter o-efj. The solid curve is the equation which appears to the left of the data points. Chapter 4. Experimental Determination of Xab(T) 95 7.79-o 7.78-| x " x 7 - 7 7 " 7.76-\ \ (a) \ \ / \ / \ / ^ .026 .027 .028 .029 .030 .031 .032 Proport ional i ty Constant "C" Proport ional i ty Constant "C" Figure 4.9: The total X2 for fits to the (a) 0.5T and (b) 1.5T data below 55K. The dashed lines are guides to the eye. Chapter 4. Experimental Determination of Aa&(T) 96 precession amplitude of the resultant signal is independent of temperature, but depen-dent upon the applied magnetic field. The asymmetry amplitude above Tc at fields of 0.5T and 1.5T were found to be A0 « 0.266(1) and A° « 0.247(1), respectively. The field dependence is primarily at t r ibuted to the finite timing resolution of the counters, which causes the observed precession amplitude to decrease as the period of the muon precession becomes comparable to the timing resolution. In the final step of this analysis, the status of the fitting parameters was then as follows: 1. S a m p l e Signal [refer to Eq. (3.39)]: Variable parameters: i) The amplitude A°sam ii) 1/A^ iii) The average internal field B iv) The initial phase 6 Fixed parameters: i) K ii) aeff fixed to Aa(, according to Eq. (4.6) 2. Background Signal [refer to Eq. (4.3)]: Variable parameters: i) The field Bbkg « Bapplied fi) CFbkg iii) The initial phase 9 (same as for sample signal) Chapter 4. Experimental Determination of Xab(T) 97 Fixed parameters: i) The amplitude, A°bkg = A°ns - A°sam Thus in the final fit of the data there were six independent parameters. Figure 4.10 shows the variation with temperature of the initial phase 0, the average field Bbkg ,th.e amplitude A^k and relaxation rate Obkg of the background precession signal, obtained from fits of the 0.5T data. As indicated in Fig. 4.10(a), the phase of the initial muon spin polarization remains nearly constant throughout the temperature scan (i.e. 89 ~ 0.05rad). This implies that there were no appreciable fluctuations in the applied field or electronics. The nearly constant field Bbkg in Fig. 4.10(b) is a further indication of a highly stable applied magnetic field. The 1.5T data is not shown because there was a significant change in the applied field after 40K. Figure 4.10(d) shows a significant drop in the relaxation rate of the background signal at higher temperatures, indicating some temperature dependence for abkg- How-ever, at lower temperatures (T < 50K) the background relaxation rate and hence the contribution of the background signal to the second moment exhibits no obvious cor-relation with temperature. This suggests that Obkg plays little role in the temperature dependence of a in Fig. 4.6. The fact that a^kg > cns suggests that the background is caused by a material with a large nuclear dipolar interaction such as Cu, or is in a region of fairly large field inhomogeneity. Figure 4.11 and Fig. 4.12 show the temperature dependence of A°sam, B and 1/X\b arising from the same fits which produced the results in Fig. 4.10. Together these parameters constitute three of the four variable parameters (the other being 0) which pertain to the signal originating from the sample. The sample amplitude A°am depicted Chapter 4. Experimental Determination of A0(,(T) 98 2 . 3 2. B75 ^ ~ N 2 - 8 5 CO C 2.825 D X I 2.8 O Q> 2. 7S -2.725 2.7 * « • • % % $ „ 4> (a) " <t> • ' 0 10 20 30 40 50 80 70 80 30 TEMPERATURE (K) 0.498 0.497S C 0.49BS -o laf 0.495S -0.495 (b) oeooo o 0 o o o j i i i_ o <t> 0 10 20 30 40 50 60 7D 80 90 TEMPERATURE (K) < 0.06 0.055 0.05 c g D.045 o> o D - 0 < o -. 0,035 c 0.03 D.025 ---a> • a n i 0 ID —i 1 1 1 1 1 1 — (c) < * • • * 1 1 1 1 1 1 1 20 30 40 50 60 70 80 TEMPERATURE (K) ->' . t 9 _ 1 CO 3 T> c o O l o D b 0 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) Figure 4.10: The temperature dependence of (a) the initial phase 0, (b) the average field Bbkg, (c) the amplitude Alkg and (d) the depolarization rate a^kg corresponding to the background signal produced by muons missing the sample, in an applied field of 0.5T. These results are taken from fits in which the total muon spin precession signal amplitude was fixed to a constant. Chapter 4. Experimental Determination of Aaj(T) 99 in Fig. 4.11(a) shows some scatter and a slight decrease at higher temperatures. The scatter in the asymmetry amplitude is not all tha t surprising considering that the data was recorded over a period of 5 days, through which time, small fluctuations in experimental conditions were unavoidable. For instance, one such experimental variation was the rate at which 4He was pumped through the cryostat. At higher temperatures (where the required cooling power is low) the amount of 4He flowing into the cryostat and the corresponding pumping rate were minimized in an effort to keep the heater voltage small to preserve the supply of 4He and to reduce thermal gradients between the thermometers and the sample. However, to maintain low temperatures a much larger flow of 4He was required. The increased density of helium atoms in the cryostat increases the probability of scattering the incoming muons before they can reach the sample, thus increasing the background signal and decreasing the magnitude of A°sam. To minimize this effect, the cryostat sample space was pumped on hard, but the choice of a specific combination of 4He-flow rate and the pumping rate was purely judgemental. This is a possible explanation for the scatter observed in Fig. 4.11(a). However, the downward trend of A°am as one increases the temperature may be purely statistical, as a similar behaviour was not observed in more recently recorded data fitted with the same procedure. Recall that since the total asymmetry amplitude was fixed, the variation of A°hkg with temperature in Fig. 4.10(c) appears as a mirror image of Fig. 4.11(a). Figure 4.11(b) shows the temperature variation of the average internal field B ex-perienced by muons implanted in the YBa2Cu306.95 sample. For comparison, the back-ground field Bbkg is also plotted in Fig. 4.11(b). In general, the field at any point in the sample is the sum of the local fields in Eq. (3.13). For all temperatures, B is less than Bbkg, but B appears to approach Bbkg at both ends of the temperature scan. In Chapter 4. Experimental Determination of Aa&(T) 100 0.245 0.24 -0.235 -0.23 0.225 -0.22 _ j | j j _ (a) <\> 4 * * <!> * 0 * * * * J __i I L 0.497 0.4368 I— ^ 0.4966 Q _J LU 0.4964 LU 0. 4962 -< L|-l 0.496 < 0.4958 0.4956 0 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) 1 A A A A A H 0 0 i i A A A 0 0 1 1 A 0 1 A 0 1 1 1 (b) A A A 0 1 1 1 ©-® © 1 0 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) Figure 4.11: The temperature dependence of (a) the muon precession amplitude A°am and (b) the average internal field B (circles), corresponding to the precession signal produced by muons hitt ing the sample, and the background field Bbkg (triangles), for an applied field of 0.5T. The total precession signal amplitude was assumed constant in the fits. Chapter 4. Experimental Determination of Xab(T) 101 the high-temperature regime the vortex cores begin to overlap with the internal field distribution approaching full penetration of the applied field. Thus it is not surprising to see the average internal field B approach Bbkg as one increases the temperature to-wards Tc. The rise in average field B at low temperatures, however, is more difficult to understand. Such an increase has also been reported in previous work by Riseman [77] and observed in more recent data taken at different fields. The cause for such behaviour is puzzling indeed. However, Fig. 4.11(b) is consistent with the time spectrum shown in Fig. 4.3 which shows a more distinct beat in the muon spin precession signal at the intermediate temperature T = 35.5K, corresponding to a greater separation between the average precession frequency of muons subjected to the internal field distribution and the average precession frequency of muons in the background field. This suggests that the increase in B at low temperatures may be due to some intrinsic phenomenon of the YBa2Cu306.95 sample itself. Figure 4.12 shows the temperature dependence of 1/A„6 (which in the phenomeno-logical London Model is directly proportional to the superfluid density ns) for the applied field of 0.5T . Since the relaxation rate aeff is assumed proportional to 1/A^6 [see Eq. (4.6)], the variation of aejf with temperature resembles the behaviour in Fig. 4.12. Figure 4.13 shows the low-temperature dependence of 1/Aj ,^ for both 0.5T and 1.5T applied fields. As shown, the presence of a linear term (i.e. 1/A^, oc T) in the low-temperature region is unmistakeable for both 0.5T and 1.5T fields, with the latter showing a weaker linear dependence on T. A fit to the low-temperature data (i.e. below 55K), with an equation of the form: gives Aa6(0) = 1451(2)A and Aa6(0) = 1496(1)A for the 0.5T and 1.5T data respectively, where the quoted uncertainties are purely statistical [87]. It should be noted that this Chapter 4. Experimental Determination of Xab(T) 102 : i CM 1 * o 50 40 30 -20 10 0 10 20 30 40 50 60 70 80 90 100 TEMPERATURE (K) Figure 4.12: The temperature dependence of 1/X2ab in a magnetic field of 0.5T, as deter-mined from fits in which the total precession signal amplitude was assumed constant. Chapter 4. Experimental Determination of Xab(T) 103 equation is purely phenomenological and cannot be extended to include the higher-temperature data. Both curves suggest that pair breaking persists at the lowest of temperatures in YBa2Cua06.95, which is inconsistent with conventional s-wave pairing of carriers. This low-temperature behaviour indicates a more complicated gap function A(k, T) characteristic of the presence of nodes in the energy gap. In Fig. 4.14 the temperature dependence of \ab at 0.5T and 1.5T is shown. The solid curves represent microwave measurements of the change in penetration depth A\ab(T) taken in zero static magnetic field by Hardy et al. [6]. For the purpose of comparison, Aa&(0) for each field is chosen to be the value obtained from fitting the ^+ S R low-temperature da ta with Eq. (4.7). Surprisingly, the microwave data shows a much better agreement with the / / + SR data at the higher magnetic field of 1.5T. There was some concern after completion of the above analysis that fixing the total asymmetry amplitude to the value above Tc may introduce systematic errors by constraining the fits. The large fluctuation in the amplitude of the muon spin precession signal originating from the sample [see Fig. 4.11(a)] was the source of such concerns. Intuitively, we expect that A°am should scale with the percentage of muons striking the sample. The fluctuations in this percentage during the actual experiment were probably not large enough to account for the large scatter in Fig. 4.11(a). If one dismisses the previous explanation for the large fluctuations in A°am, it is worth investigating this mat ter further. Since A°sam is not expected to change significantly over the temperature scan, the da ta was refitted first by designating A°sam and A\kg as variable parameters. As in the previous analysis, cre// was assumed to be proportional to 1/A^6 through Eq. (4.6). The proportionality constant C was determined to be 0.0250(10) and 0.0243(10)/um2^s_1 for the 0.5T and 1.5T data respectively. The variable parameters pertaining to the Chapter 4. Experimental Determination of Xab{T) 104 10 20 30 40 TEMPERATURE (K) 50 60 Figure 4.13: The temperature dependence of 1/A^ in magnetic fields of 0.5T and 1.5T, as determined from fits in which the total precession signal ampli tude was assumed constant. The solid lines are fits to Eq. (4.7). Chapter 4. Experimental Determination of Xab(T) 105 o < 190Q 1800 1700 r < 1600 -1500 I 400 ----I I (a) 0.5T CD © CD / -CD ^ ^ ^ I I ! CD i <D l CD i — CD ---10 20 30 40 50 60 TEMPERATURE (K) o < O 10 20 30 40 TEMPERATURE (K) 50 60 Figure 4.14: The temperature dependence of Xab at (a) 0.5T and (b) 1.5T. The solid lines show the microwave measurements of AA0f,(T) in zero field from Ref. [6], assuming Xab(0) = 1451 and 1496A in (a) and (b) respectively. Chapter 4. Experimental Determination of Xab(T) 106 background precession signal varied with temperature according to Fig. 4.15. Com-paring with Fig. 4.10, the phase 9 shifts down ~ 0.005rad, while Bbkg shifts upward ~ 0.05mT. The degree of fluctuation in both these parameters appears similar to that of the previous analysis, so again it seems apparent that there were negligible fluctuations in the applied field. The amplitude A°bkg and the relaxation rate a^g [see Fig. 4.15(c) and Fig. 4.15(d)] show almost no change from the results depicted in Fig. 4.10. Even the size of the statistical error bars are comparable. These results indicate that the fitting program is capable of clearly separating the unwanted background signal from the sample signal. Figure 4.16(a) shows the temperature dependence of the amplitude A°am corre-sponding to the muon spin precession signal originating from the sample. The down-ward trend with increasing T appears slightly more prominent than in Fig. 4.11(a). The temperature dependence of B in Fig. 4.16(b) is significantly different from that in the previous analysis. The average field B is greater than Bbkg at the lowest of temperatures and does not dip as far below Bbkg as in Fig. 4.11(b) for temperatures beyond this. At the high-temperature end in Fig. 4.11(b), B recovers to approximately the same value as in Fig. 4.11(b). Again the rise in B at low temperatures is surprising. —* —* It is possible that this is an effect due to a-b anisotropy. The presence of a-b anisotropy would distort the vortex lattice into isoceles triangles as shown in Fig. 3.9. If this lattice were to be modelled by one consisting of equilateral triangles as assumed in our analysis, then there would be some error in the determination of the average field B. This would be a greater problem at low temperatures where the cores are further apart and errors in spectral weighting are more pronounced. The low-temperature dependence of 1/A^6 is shown in Fig. 4.17. Surprisingly, the scatter in the data points is not significantly greater than in Fig. 4.13. Noticeably Chapter 4. Experimental Determination of A0(,(T) 107 10 20 30 40 50 BO 70 80 3D TEMPERATURE (K) 0.497S loo 5 0.496 Xi 0.49SS 0e©oo o 0 © (b) A " 0 10 20 30 40 SD 60 70 80 SO TEMPERATURE (K) 0.06 0. 055 0.05 § 0. 045 2 0.04 o o •° 0. 035 0.03 0.025 > • • i D 10 (c) < * * * * * o 6 lj> 0 1 1 t 1 1 1 1 20 30 40 50 60 70 80 TEMPERATURE (K) • >' . f 9 T CO 3 O J* o o A b 0 0.4 0.3S 0.3 0. 2S 0.2 0.15 0.1 0.05 0 0 10 20 3D 40 50 60 70 TEMPERATURE (K) Figure 4.15: The temperature dependence of (a) the initial phase 6, (b) the average field Bi,kg, (c) the amplitude A\hg and (d) the depolarization rate a^kg corresponding to the background signal produced by muons missing the sample, in an applied field of 0.5T. These results are taken from fits in which A°am and A°bkg were free to vary. Chapter 4. Experimental Determination of Xab(T) 108 0.245 0.24 0.235 0.23 0.225 0.22 0.215 H j _ (a) C) () () Q J L J I I 0.4974 0.4372 t 0.497 Q Cj 0.4968 U-i 0.4966 O < UJ 0.4964 > < 0.4962 h 0.496 (b) . AA A * A * 0 A 4i $ * * J I 1 L D 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) A I ©• © 0 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) Figure 4.16: The temperature dependence of (a) the muon precession ampli tude A°sam and (b) the average internal field B (circles), corresponding to the precession signal produced by muons hitting the sample, and the background field Bbkg (triangles), for an applied field of 0.5T. A°am and A°hkg were both independently variable parameters in the fits. Chapter 4. Experimental Determination of A0t(T) 109 different however, is an increase in the linear term (see Table 4.1). Furthermore, fits to Eq. (4.6) yield Ao6(0) = 1350(2)A and Aafc(0) = 1437(1)A for the 0.5T and 1.5T data, respectively. A comparison to the microwave measurements of Hardy et a/., assuming these values for Aaf,(0) is shown in Fig. 4.18. There appears to be even less agreement at 0.5T than previously noted in Fig. 4.14(a). However, the agreement at 1.5T in Fig. 4.18(b) is comparable to that in Fig. 4.14(b) despite the significant difference in Aofc(0). As a final step in the analysis, the data was refitted with the amplitude A°am fixed to the average value of the data below 55K in Fig. 4.16(a). This results in a noticeable reduction in the scatter for the parameters A°hkg and a^kg (see Fig. 4.19). Fixing A°am in this way significantly shifts the data points above 40K. This is not surprising since A°am was fixed to the low-temperature average. The phase 9 shows a slight decrease at high temperatures [see Fig. 4.19(a)] and B levels off above 40K [see Fig. 4.20(b)]. These results suggest that fixing A°am to the low-temperature average reduces the scatter in the low-temperature data, but it is not yet clear whether or not we are introducing non-physical deviations in the high-temperature region. The reduction in scatter is most noticeable in Fig. 4.21 which shows the temperature dependence of l/\2ab. From Eq. (4.6) we find Aa6(0) = 1362(2)A and Aa6(0) = 1445(1)A for the 0.5T and 1.5T data , respectively. A plot of the temperature dependence of 1/Aj^ over the full temperature scan is shown in Fig. 4.22. The two fields appear to converge well before Tc, but the crossover is difficult to determine. As shown in Fig. 4.23, there is improved agreement between the microwave measurements and the 0.5T / / + SR data, while the agreement with the 1.5T data is comparable to that of the previous two fitting methods. The total asymmetry amplitude of the muon spin precession signal as determined from all three fitting procedures is shown in Fig. 4.24. It appears as Chapter 4. Experimental Determination of Aa&(T) 110 E =1 CM I o E CM I XI o r < 10 20 30 40 50 TEMPERATURE (K) 60 Figure 4.17: The temperature dependence of 1/A^6 in magnetic fields of 0.5T and 1.5T, as determined from fits in which A°am and Alkg were variable parameters. The solid lines are fits to Eq. (4.7). Chapter 4. Experimental Determination of Xab(T) 111 °< r< 1 oou 1800 1700 1600 1500 1400 1 onn _ -<$>, I I (a) 0.5T © 4> CD ^ ^ ^ ^ l CD [ CD I 0 0 -. / -10 20 30 40 50 60 TEMPERATURE (K) o < r< 10 20 30 40 50 TEMPERATURE (K) 60 Figure 4.18: The temperature dependence of Xab at (a) 0.5T and (b) 1.5T. The solid lines show the microwave measurements of AA„b(T) in zero field from Ref.[6], assuming Xab(O) — 1350 and 1437A in (a) and (b), respectively. Chapter 4. Experimental Determination of \ab(T) 112 2.3 2.875 CO '•" c D 2.825 D Q i 2.75 2. 725 2.7 2.775 4 0 , 4 * ^ $ <t> .J) (a) 20 30 40 50 60 70 80 90 TEMPERATURE (K) 0. 497S - ^ 0.437 g 0.4365 o> 8 0 .496 I CD 0.49SS -O c p o o 0 0 o 1 1 1 0 o o ( b ) • o o G i i i 0 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) 0.06 0.055 0.05 0.045 0.04 0.035 0.03 0.025 0.02 --• n o ' <D 1 1 1 i 1 1 (c) ; 0 © T CO 3 T3 C 3 O CT> - x O o b 0.35 0.3 0.25 0.2 0. IS 0.1 0.05 0 r r r w : * • * - i - i - i - i -*^ 0 10 20 30 40 50 60 70 60 90 TEMPERATURE (K) 0 tO 20 30 40 50 60 70 80 90 TEMPERATURE (K) Figure 4.19: The temperature dependence of (a) the initial phase 6, (b) the average field Bbkg, (c) the amplitude Alkg and (d) the depolarization ra te &bkg corresponding to the background signal produced by muons missing the sample, in an applied field of 0.5T, from fits where A°am is assumed constant. Chapter 4. Experimental Determination of Aa(,(T) 113 < 0.24 0.2375 0.235 0.2325 0.23 0.2275 0.225 0.2225 0.22 i i 1 1 1 r (a) -ooooo oo o o o o o o oao-J I I I I I I L 0.4974 0.4972 ^ 0.497 Q _l LU 0.4968 y 0.4966 < LU 0.4964 > < 0.4962 0.496 0 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) 1 1 A A A A A A -1 1 ! A 0 1 A 0) A O ! A * i i (b) A A l 1 i A* A © axD. i D 10 20 30 40 50 60 70 80 90 TEMPERATURE (K) Figure 4.20: The temperature dependence of (a) the muon precession amplitude A°am and (b) the average internal field B (circles), corresponding to the precession signal produced by muons hitt ing the sample, and the background field Bbkg (triangles), for an applied field of 0.5T. Chapter 4. Experimental Determination of \ab(T) 114 though one is justified in fixing the total asymmetry amplitude, as the average values are comparable. The results from all three types of analysis are summarized in Table 4.1. Methods (ii) and (iii) give comparable results, but differ substantially from method (i). The difference appears to be related to the proportionality constant C of Eq. (4.6). As C increases, so does Ao(,(0). It should be noted that for method (i) in Table 4.1, the total asymmetry amplitude was fixed prior to the determination of C. This may in fact be the most significant difference between method (i) and the other fitting procedures, in which C was de-termined before fixing any additional parameters. To see if this is the case, the 0.5T data was refit, by first determining the proportionality constant C and then fixing the total asymmetry amplitude to the average value for the data below 55K (see method (iv) in Table 4.1). Remarkably, the total asymmetry amplitude was found to be the same as in method (i) (i.e. A0 » 0.2650). The linear coefficient a and the quadratic coefficient 0 [determined by fitting the low-temperature data to Eq. (4.7)] are virtually the same for methods (i) and (iv), but the values obtained for Aft5(0) are very different. Moreover, the value of Aaf,(0) from method (iv) is comparable to (ii) and (iii). All of this implies that Xab(0) is significantly influenced by changes in C, but is little affected by the manner in which the amplitude of the precession signal is treated in the fitting procedure. Also, the deviations in the linear term from one method to the next are likely not significant enough to suggest that there is any difference in the behaviour of Xab(T) at low temperatures. Chapter 4. Experimental Determination of Aa(,(T) 115 0 20 30 40 50 TEMPERATURE (K) 60 Figure 4.21: The temperature dependence of 1/A^6 in magnetic fields of 0.5T and 1.5T, as determined from fits in which A°am was assumed to be constant. The solid lines are fits to Eq. (4.7). Chapter 4. Experimental Determination of A0j(T) 116 E 3. CM I o 60 50 40 30 20 10 -0 10 20 30 40 50 60 70 80 TEMPERATURE (K) 90 100 Figure 4.22: The temperature dependence of 1/A^ in magnetic fields of 0.5T and 1.5T, as determined from fits in which A°am was assumed to be constant. Chapter 4. Experimental Determination of Xab(T) 117 1750 1700 1650 1600 ° < , 1550 -4? 1500 1450 1400 1350 1300 r< ---1 1 (a) 0.5T a> CD / CD . ^ CD ^ ^ 1 1 I © > -1 CD y 1 CD / O --10 20 30 40 50 60 TEMPERATURE (K) o < r< 10 20 30 40 50 TEMPERATURE (K) 60 Figure 4.23: The temperature dependence of Aa& at (a) 0.5T and (b) 1.5T. The solid lines show the microwave measurements of A\ab(T) in zero field from Ref.[6], assuming Aa;,(0) = 1362 and 1445A in (a) and (b) respectively. Chapter 4. Experimental Determination of Xab{T) 118 0.285 0.28 0.275 0.27 0.265 0.26 0.255 0.25 0.245 1 t f -ooooo oo o 1 1 1 o 1— o '" 0 1 1 ( a ) • o o o a> i i UJ Q 0-< >-t-UJ >-in < < i— O 0.28 0.275 0.27 0.265 0.26 0.255 0.25 T 1 I 1 1 P 1 1 1 + 1 1 —1 ( b ) • -• t i < > i i i 0.285 0.28 0.275 0.27 0.265 0.26 0.255 0.25 0.245 "T 1 1 1 1 1 1 T" (c) CD CD CD J I 1^  I I I I 1_ 0 10 20 30 40 50 60 70 80 TEMPERATURE (K) 90 Figure 4.24: The temperature dependence of the total asymmetry amplitude at 0.5T for three different fitting procedures: (a) (A°sam 4- A°bkg) is fixed to a constant value; (b) A°am and A°bkg are independent parameters; (c) A°am is a fixed parameter. Chapter 4. Experimental Determination of Xab(T) 119 Fitting Procedure Method (i): (A am is "fixed before" determining C Method (ii): A°sam a n d A°hkg are "free" Method (iii): A°sam is "fixed-Method (iv): (A°sam + A°bkg) is "fixed after" determining C Applied Field (T) 0.5 1.5 0.5 1.5 0.5 1.5 0.5 "C" from Eq. (3.6) 0.0293 0.0258 0.0250 0.0243 0.0250 0.0243 0.0250 Aoi(0) (A) 1451(2) 1496(1) 1350(2) 1437(1) 1362(2) 1445(1) 1347(2) a (K-1) 7.2(1) x lO - 3 3.4(5) x lO"3 6.4(1) x lO"3 4.4(1) x lO"3 6.5(1) x lO"3 3.7(1) x lO"3 7.7(3) x lO"3 0 (K-2) 0 4.5(8) x 10~5 2.6(1) x lO - 5 3.5(1) x lO"5 4.8(2) x lO"6 4.2(1) x lO"4 0 Table 4.1: Comparison of the fitting procedures. Chapter 5 Conclus ion The observation of a linear temperature dependence below 50K for 1/A^6 is in agree-ment with the microwave cavity measurements of Hardy et al. on similar YBa2Cu306.95 crystals in zero applied field [6]. Both experiments provide further evidence for uncon-ventional pairing of carriers in the superconducting state. These results of course contradict previous / / + SR studies, which found a much weaker low-temperature be-haviour for \ab [9]. The likelihood of impurity scattering playing a role in suppressing Xab in such a way as to simulate conventional s-wave behaviour has been made more plausible by recent measurements on Zn-doped crystals [84,88]. These measurements show a distinct weakening of the linear term at low temperatures due to the added Zn impurity. Thus it is possible that the presence of impurities, as well as a lack of good low-temperature data may have lead to a misinterpretation in some of the previous / i + SR experiments. The difficulties in analyzing / /+SR data of this nature have been addressed as much as possible in this study. The large number of variable parameters requires one to make some plausible assumptions in the fitting procedure. Fortunately, all variations of the analysis considered in this report arrive at the same conclusion regarding the behaviour of \ab(T) at low temperatures; namely, a strong linear term exists. Furthermore, the strength of the linear term is comparable for all forms of analysis considered. This implies that the observed low-temperature linear dependence of 1/A^6 is not an artificial manifestation of the fitting procedure itself. This notion is further supported by the 120 Chapter 5. Conclusion 121 obseravtion of a linear term in the single gaussian fits, which provide a crude estimate of the second moment. The weakening of the linear term at 1.5T (or conversely, the strengthening of the linear term at 0.5T) was surprising indeed. The magnetic penetration depth is not expected to be field dependent in this low-field regime. Theoretically there is no low-field limit associated with the field distribution used to model the vortex lattice. Eq. (3.9) is simply an extension of the London model which has no low-field limit. One possible explanation for the observed field dependence is quasiparticle scatter-ing off of the vortex cores, which we know to be static, as evidenced by the field-shifted results of Fig. 4.5. One can imagine this effect to be enhanced at higher magnetic fields, where the flux-line density is greater in the sample. A scattering process of this nature may be similar to the impurity scattering which appears to weaken the linear term in the Zn-doped samples. It is possible that the observed low-temperature field dependence for Aaj, is some-how linked to the a-b anisotropy in the penetration depth, not considered here. It is important to stress that none of the previous / / + SR studies included a-b anisotropy in determining the temperature dependence of Aa(,, either. Consequently, it cannot be held accountable for the observation of a linear term in the present study. Another puzzling observation comes from the comparison of the temperature de-pendence of \ab with that obtained from the microwave cavity measurements. The 1.5T data agrees well with the microwave results for all forms of the analysis. On the other hand, the 0.5T data shows poor agreement with the microwave measurements. The better agreement with the higher-field / i + SR data is surprising since the microwave measurements were performed in zero static magnetic field. However, there are some questions as to whether the two types of measurements can be compared at this level Chapter 5. Conclusion 122 due to the very different nature of the two methods. In the microwave studies the mea-sured penetration depth pertains to the length scale over which very weak shielding currents flow around the perimeter of the sample. In the /u+SR studies one is measur-ing the penetration depth associated with supercurrents circulating around the vortex cores in the bulk of the sample. Finally, something must be said about the uncertainty in the / / + SR measurements. This study gives Aa(,(0) in the range 1347 - 1451A and 1437 - 1496A depending on the analysis, for the 0.5T and 1.5T fields, respectively. However, the errors in these results are difficult to determine. The systematic errors are much larger than the statistical errors quoted in Table 4.1 and those which appear on the graphs throughout this report. Thermal and magnetic field fluctuations during the experiment are difficult to assess, but these uncertainties are likely negligible compared to those introduced in the fitting procedure. Although there is some question regarding the accuracy of the Aaj(0) values obtained, there is exciting new qualitative information to be obtained from this /x+SR study. Namely, evidence for unconventional pairing of carriers in the superconducting state of YBa2Cus06.95 and the possible existence of a low-temperature field dependence for Xab. B i b l i o g r a p h y [1] P. 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Keller, Earlier and Recent Aspects of Superconductivity, edited by J.G. Bednorz and K.A. Midler, 222 (Springer-Verlag, Berlin, 1990). [74] A.D. Sidorenko, V.P. Smilga and V.I. Fesenko, Hyperfine Interactions 6 3 , 49 (1990). [75] V. Fesenko, V. Gorbunov, A. Sidorenko and V. Smilga, Physica C 2 1 1 , 343 (1993). [76] E.H. Brandt , Journal of Low Temp. Phys. 26, 709 (1977) [77] T.M. Risemann, Ph.D. thesis, University of British Columbia (1993) [78] S.L. Thiemann, Z. Radovic and V.G. Kogan, Phys. Rev. B 39 , 11406 (1989). [79] A. Schenck, Muon Spin Rotation Spectroscopy: Principles and Applications in Solid State Physics (Adam Hilger Ltd., 1985). [80] S.F.J. Cox, J. Phys. C 20, (1987). [81] E.H. Brandt and A. Seeger, Adv. in Phys. 35 , 189 (1986). [82] P. Monthoux, A.V. Balatsky and D. Pines, Phys. Rev. B 46, 14803 (1992). [83] K. Ishida, Y. Kitaoka, N. Ogata, T. Kamino, K. Asayama, J.R. Cooper and N. Athanassopoulu, J. Phys. Soc. Jpn. 62, 2803 (1993). [84] D.A. Bonn et al. Oxygen Vacancies, Zinc Impurities and the Intrinsic Microwave Loss ofYBa.2Cu3Oe.95, submitted to Phys. Rev. B (1993). [85] R.X. Liang, P. Dosanjh, D.A. Bonn, D.J. Barr, J .F . Carolan and W.N. Hardy, Physica C 195, 51 (1992). [86] J .W. Schneider, R.F. Kiefl, K.H. Chow, S. Johnston, J .E. Sonier, T.L. Estle, B. Hitti, R.L. Lichti, S.H. Connell, J .P .F . Sellschop, C.G. Smallman, T.R. Anthony and W.F. Banholzer, Phys. Rev. Lett. 7 1 , 557 (1993). [87] J.E. Sonier, R.F. Kiefl, J.H. Brewer, D.A. Bonn, J .F . Carolan, K.H. Chow, P. Dosanjh, W.N. Hardy, Ruixing Liang, W.A. MacFarlane, P. Mendels, G.D. Morris, T.M. Riseman and J .W. Schneider, Phys. Rev. Lett. 72 , 744 (1994). [88] Recent / / + SR measurements of the temperature dependence of Aa6 in YBa2(Cuo.997Zn0.oo3)306.95, in an applied field of 0.2T, show a distinct weakening of the linear term in agreement with microwave cavity measurements performed on the same sample., unpublished (1994). A p p e n d i x A S o m e R e m a r k s A b o u t T h e F i t t ing P r o g r a m The computer time allocated to calculating B(r) of Eq. (3.9), is significant enough that it is practical to avoid the large sum over reciprocal lattice vectors K for each iteration in the ^ -min imiza t ion procedure. Consequently, a Taylor series expansion around an initial value of the magnetic penetration depth A0 was employed in the actual fitting program: 1 1 / 1 1 \ dB (r, ±) so that , B(r,js) is determined by expanding about the initial point (r,-^). In this way, B(f) is calculated by summing over the reciprocal lattice vectors K [in Eq. (3.9)], only once for an initial set of parameters -A- and B0, where B0 is the initial value of the average field. It is not necessary to expand about B0. This is because changes in the average field merely shift the field distribution along the field (or frequency) axis. In the fitting process, field shifts in excess of a conservative value of ~ 0.00073T were not permit ted before the program was stopped, the initial parameters changed, and the data refitted. Table A.l shows the accuracy of the Taylor series assuming an average magnetic field 0.5T and K = 68. The error in using Eq. (A. l ) , expressed as a percentage of the exact calculation of B(r) using Eq. (3.9), is shown for B(r) calculated at the vortex 128 Appendix A. Some Remarks About The Fitting Program 129 1/A2 (^m- 2 ) 50 50 50 50 50 50 50 50 50 50 50 50 1/A2 (^m- 2 ) 50 49 48 47 46 45 44 43 30 20 10 0 % Error in B(f) at the vortex core. 0.00 0.16 0.31 0.47 0.62 0.78 0.93 1.08 3.01 4.43 5.78 7.07 % Error in B(r) at the saddle -point. 0.00 0.14 0.27 0.40 0.53 0.67 0.80 0.93 2.65 3.96 5.25 6.54 Table A. l : Accuracy of the Taylor series expansion used in the fitting program. Appendix A. Some Remarks About The Fitting Program 130 core and at a saddle point. The results show that even large changes in -^ give a good approximation for B(r). In a typical fit, B(r) is calculated in excess of 150, 000 times, so that using a Taylor series greatly diminishes the time required to fit. 

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